Abstract group 81000.t: $C_{15}\wr S_3:C_4$

• Label: 81000.t
• Order: \( 2^{3} \cdot 3^{4} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{5} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_{15}\wr S_3:C_4$
Order:	\(81000\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5^{3} \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_{15}^2.(C_{12}\times S_3^2)\times F_5$, of order \(1944000\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5^{3} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4, $C_5$ x 3
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	5	6	9	10	12	15	18	30	45
Elements	1	395	476	13500	124	7660	900	1980	27000	5024	4500	15840	3600	81000
Conjugacy classes	1	3	6	4	9	14	1	7	4	164	1	56	2	272
Divisions	1	3	6	2	9	14	1	7	2	86	1	28	1	161
Autjugacy classes	1	3	4	4	5	8	1	4	4	24	1	8	1	68

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	8	8	8	16	3	120	0	109	0	272
Irr. rational chars.	4	8	5	14	3	13	1	63	50	161

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$432$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{12}=c^{15}=d^{15}=e^{15}=[a,c]=[a,e]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2,3,5,6,10,14,8,11,13)(4,7,12,15,9)(16,17)(18,19,21)(20,23)(22,24), (2,4,8,12)(3,6,11,14)(5,9,10,15)(7,13)(17,18,20,19,22,21)(23,24)\rangle$
Transitive group:	45T956				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{15}\wr S_3)$ $\,\rtimes\,$ $C_4$	$(C_{15}^3.C_{12})$ $\,\rtimes\,$ $C_2$	$(C_{15}\wr C_3:C_4)$ $\,\rtimes\,$ $C_2$	$C_5$ $\,\rtimes\,$ $(C_{15}^2:C_6.D_6)$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{15}^2$ . $(D_5.S_3^2)$	$(C_{15}\wr S_3:C_2)$ . $C_2$	$(C_3^3\times C_5^2:D_5)$ . $D_6$	$C_3^2$ . $(C_5^3:C_6.D_6)$	all 16

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 425944 subgroups in 2312 conjugacy classes, 38 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{15}\wr S_3:C_4$
Commutator:	$G' \simeq$ $C_{15}\wr C_3$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_5^3:C_6.D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{15}^3$	$G/\operatorname{Fit} \simeq$ $C_4\times S_3$
Radical:	$R \simeq$ $C_{15}\wr S_3:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^2\times C_{15}$	$G/\operatorname{soc} \simeq$ $C_6.S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 81000: $C_{15}\wr S_3:C_4$

Order 40500: $C_{15}\wr S_3:C_2$, $C_{15}^3.C_{12}$, $C_{15}\wr C_3:C_4$

Order 27000: $C_{15}^3.C_2^2.C_2$, $C_{15}^2:(S_3\times F_5)$

Order 20250: $C_{15}\wr S_3$, $(C_3\times C_{15}^2):D_{15}$, $C_{15}\wr C_3:C_2$

Order 16200: $C_{15}^2:C_6.D_6$

Order 13500: $C_{15}^3.C_4$, $C_3^2\times C_5^3:D_6$, $(C_5\times C_{15}^2).C_{12}$, $(C_5\times C_{15}^2):D_6$, $(C_5\times C_{15}^2):C_{12}$, $C_5^2:C_{45}:C_{12}$, $C_{15}^2:C_{15}:C_4$

Order 10125: $C_{15}\wr C_3$

Order 9000: $C_5^3.(C_6\times S_3).C_2$ x 4, $C_5^3.C_6^2.C_2$, $C_5^3:C_6.D_6$

Order 8100: $(C_3\times C_{15}^2):C_{12}$, $(C_3\times C_{15}^2):D_6$, $C_{15}^2:C_6.S_3$

Order 6750: $C_3^2\times C_5^3:S_3$, $C_{15}^2\times D_{15}$, $C_3^3\times C_5^2:D_5$, $C_5\times C_{15}^2:S_3$, $C_{15}^2:D_{15}$, $(C_5\times C_{15}^2).C_6$, $(C_5\times C_{15}^2):C_6$

Order 5400: $C_{15}^2.C_6.C_2^2$ x 6, $C_{15}^2.C_6.C_2^2$, $C_{15}^2:(C_4\times S_3)$

Order 4500: $C_5^3:(C_3^2:C_4)$ x 5, $C_3\times C_5^3:D_6$ x 4, $C_3\times C_5^3:(C_3:C_4)$ x 3, $C_5^3:(C_3^2:C_4)$, $C_3^2\times D_5\times C_5:D_5$, $C_5^3:(C_3\times C_3:C_4)$, $C_3\times C_5^3:D_6$, $(C_5^2\times C_{15}):C_{12}$, $C_5^2:C_{45}:C_4$, $C_5^3:C_3^2:C_4$

Order 4050: $(C_3\times C_{15}^2):S_3$ x 2, $(C_3\times C_{15}^2):C_6$

Order 3375: $C_{15}^3$, $C_{15}^2.C_{15}$, $C_{15}^2:C_{15}$

Order 3240: $C_3^3:(S_3\times F_5)$

Order 3000: $C_5^3:(C_2\times C_3:C_4)$ x 4, $C_5^3:(C_4\times S_3)$, $C_5^3:(C_4\times S_3)$

Order 2700: $C_5^2:(C_3^3:C_4)$ x 9, $C_5^2:(C_3\times C_3^2:C_4)$ x 6, $C_3^2\times D_{15}:D_5$ x 6, $C_5^2:(C_3\times C_3^2:C_4)$, $C_3^2\times S_3\times C_5:D_5$, $C_{15}^2:C_{12}$, $C_{15}^2:D_6$, $C_5^2:C_9:C_{12}$, $C_{15}^2:C_3:C_4$

Order 2250: $C_3^2\times C_5^2:D_5$ x 5, $C_5\times C_{15}\times D_{15}$ x 4, $C_3\times C_5^3:S_3$ x 4, $C_3^2\times C_5\times C_5:D_5$, $C_{15}^2\times D_5$, $C_3\times C_5\wr S_3$, $C_3\times C_5^2:D_{15}$, $C_5^3:C_{18}$, $C_3\times C_5^3:C_6$

Order 2025: $C_{15}^2.C_3^2$

Order 1800: $(C_{15}\times D_{15}):C_4$ x 24, $C_{15}^2:(C_2\times C_4)$ x 6, $S_3\times C_{15}:F_5$ x 4, $C_2\times C_{15}^2:C_4$, $C_{15}:F_5:S_3$

Order 1620: $C_3\wr C_3:F_5$, $(C_3^2\times C_{15}):C_{12}$, $C_3\wr S_3\times D_5$

Order 1500: $(C_5\times C_{15}):F_5$ x 5, $C_{15}:D_5^2$ x 4, $(C_5\times C_{15}):F_5$ x 4, $C_{15}:D_5^2$, $C_5^3:C_{12}$, $C_5^3:D_6$, $C_5^3:C_{12}$, $C_5\wr C_3:C_4$

Order 1350: $C_{15}^2:C_6$ x 12, $C_{15}^2:C_6$ x 9, $C_{15}^2:S_3$ x 2, $C_{15}^2:C_6$, $C_{15}^2.C_6$, $C_{15}^2:C_6$, $S_3\times C_{15}^2$

Order 1125: $C_5\times C_{15}^2$ x 5, $C_5^3:C_3^2$, $C_5^2:C_{45}$

Order 1080: $S_3\times C_3^2:F_5$ x 7, $C_3^2:(S_3\times F_5)$

Order 1000: $C_5^3:(C_2\times C_4)$

Order 900: $C_{15}^2:C_4$ x 86, $C_{15}^2:C_2^2$ x 24, $C_{15}^2:C_4$ x 24, $C_3^2\times D_5^2$ x 6, $C_{15}^2:C_4$ x 6, $C_{15}^2:C_2^2$ x 4, $C_{15}^2:C_4$ x 4, $C_3\times C_5^2:D_6$, $(C_5\times C_{15}):C_{12}$, $C_5^2:C_3^2:C_4$, $C_5^2:C_9:C_4$, $C_5^2:C_6^2$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 750: $C_5^3:C_6$ x 5, $C_5^2:C_{30}$ x 4, $C_5^2:C_{30}$ x 4, $C_5^2:D_{15}$, $C_5^2\times D_{15}$, $C_5^3:C_6$, $C_5^2:D_{15}$, $C_5\wr S_3$

Order 675: $C_3\times C_{15}^2$ x 9, $C_{15}^2:C_3$, $C_{15}^2.C_3$

Order 648: $C_3^3:(C_4\times S_3)$

Order 600: $D_5^2.S_3$ x 24, $D_{15}:F_5$ x 6, $C_{30}:F_5$ x 4, $S_3\times C_5:F_5$, $C_5^2:(C_4\times S_3)$

Order 540: $C_3^3:F_5$ x 9, $C_{15}:C_6^2$ x 7, $C_3^3:F_5$ x 7, $\He_3:F_5$ x 2, $\He_3:D_{10}$, $C_{45}:C_{12}$

Order 500: $C_5:D_5^2$, $C_5^3:C_4$, $C_5^3:C_4$

Order 450: $C_{15}^2:C_2$ x 86, $C_{15}\times D_{15}$ x 48, $C_{15}:C_{30}$ x 12, $C_{15}:D_{15}$ x 4, $C_{15}:C_{30}$ x 4, $C_3\times C_5^2:S_3$ x 2, $C_{15}\times C_{30}$, $C_3\times C_5^2:C_6$, $C_5^2:C_{18}$

Order 405: $C_3^3:C_{15}$

Order 375: $C_5^2\times C_{15}$ x 5, $C_5\wr C_3$

Order 360: $D_5.S_3^2$ x 29, $C_2\times C_3^2:F_5$ x 7

Order 324: $C_3^3:D_6$, $C_3\wr C_3:C_4$, $C_3^3:C_{12}$

Order 300: $C_{15}:F_5$ x 89, $C_3\times D_5^2$ x 24, $C_{15}:F_5$ x 24, $C_{15}:D_{10}$ x 6, $C_{15}:F_5$ x 6, $C_{15}:D_{10}$ x 4, $C_{15}:D_{10}$, $C_{15}:F_5$, $C_5^2:D_6$, $C_5^2:C_{12}$, $C_5^2:C_3:C_4$

Order 270: $D_5\times C_3^3$ x 9, $C_3^2\times D_{15}$ x 7, $C_3^2:C_{30}$ x 7, $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 250: $C_5^3:C_2$, $C_5^2:C_{10}$, $D_5\times C_5^2$

Order 225: $C_{15}^2$ x 85, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $C_6.S_3^2$, $C_6.S_3^2$

Order 200: $D_5:F_5$ x 6, $C_{10}:F_5$

Order 180: $C_3^2:F_5$ x 93, $C_{15}:D_6$ x 29, $C_{15}:C_{12}$ x 29, $C_3^2\times D_{10}$ x 7, $C_{45}:C_4$

Order 162: $C_3\wr S_3$ x 2, $C_3^3:C_6$

Order 150: $C_{15}:D_5$ x 89, $D_5\times C_{15}$ x 48, $C_5\times D_{15}$ x 12, $C_5\times C_{30}$ x 4, $C_5^2:S_3$ x 2, $C_5^2:C_6$, $C_5:D_{15}$, $S_3\times C_5^2$

Order 135: $C_3^2\times C_{15}$ x 9, $C_9:C_{15}$, $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $C_{30}:C_4$ x 28, $S_3\times F_5$ x 8

Order 108: $C_3^2:C_{12}$ x 2, $C_9:C_{12}$, $C_3^2\times D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:D_6$

Order 100: $C_5:F_5$ x 10, $D_5^2$ x 6, $C_5:F_5$ x 6, $C_5:D_{10}$

Order 90: $C_3^2\times D_5$ x 93, $C_3\times D_{15}$ x 29, $S_3\times C_{15}$ x 29, $C_3\times C_{30}$ x 7, $C_9\times D_5$

Order 81: $C_3\wr C_3$

Order 75: $C_5\times C_{15}$ x 85, $C_5^2:C_3$

Order 72: $C_6.D_6$ x 5, $C_6.D_6$

Order 60: $C_{15}:C_4$ x 114, $C_3\times D_{10}$ x 28, $S_3\times D_5$ x 8, $C_3\times F_5$ x 8

Order 54: $C_3^2:S_3$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 50: $C_5\times D_5$ x 12, $C_5:D_5$ x 10, $C_5\times C_{10}$

Order 45: $C_3\times C_{15}$ x 86, $C_{45}$

Order 40: $C_2\times F_5$ x 7

Order 36: $C_3^2:C_4$ x 7, $C_3:C_{12}$ x 5, $C_6\times S_3$ x 5, $C_6^2$, $C_9:C_4$

Order 30: $C_3\times D_5$ x 114, $C_{30}$ x 28, $D_{15}$ x 8, $C_5\times S_3$ x 8

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 25: $C_5^2$ x 9

Order 24: $C_6:C_4$ x 4, $C_4\times S_3$ x 2

Order 20: $F_5$ x 16, $D_{10}$ x 7

Order 18: $C_3\times S_3$ x 10, $C_3\times C_6$ x 8, $C_{18}$

Order 15: $C_{15}$ x 86

Order 12: $C_3:C_4$ x 10, $C_2\times C_6$ x 4, $D_6$ x 2, $C_{12}$ x 2

Order 10: $D_5$ x 16, $C_{10}$ x 7

Order 9: $C_3^2$ x 6, $C_9$

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 14, $S_3$ x 4

Order 5: $C_5$ x 9

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 6

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 81000: $C_{15}\wr S_3:C_4$

Order 40500: $C_{15}\wr S_3:C_2$, $C_{15}^3.C_{12}$, $C_{15}\wr C_3:C_4$

Order 27000: $C_{15}^3.C_2^2.C_2$, $C_{15}^2:(S_3\times F_5)$

Order 20250: $C_{15}\wr S_3$, $(C_3\times C_{15}^2):D_{15}$, $C_{15}\wr C_3:C_2$

Order 16200: $C_{15}^2:C_6.D_6$

Order 13500: $C_{15}^3.C_4$, $C_3^2\times C_5^3:D_6$, $(C_5\times C_{15}^2).C_{12}$, $(C_5\times C_{15}^2):D_6$, $(C_5\times C_{15}^2):C_{12}$, $C_5^2:C_{45}:C_{12}$, $C_{15}^2:C_{15}:C_4$

Order 10125: $C_{15}\wr C_3$

Order 9000: $C_5^3.(C_6\times S_3).C_2$ x 2, $C_5^3.C_6^2.C_2$, $C_5^3:C_6.D_6$

Order 8100: $(C_3\times C_{15}^2):C_{12}$, $(C_3\times C_{15}^2):D_6$, $C_{15}^2:C_6.S_3$

Order 6750: $C_3^2\times C_5^3:S_3$, $C_{15}^2\times D_{15}$, $C_3^3\times C_5^2:D_5$, $C_5\times C_{15}^2:S_3$, $C_{15}^2:D_{15}$, $(C_5\times C_{15}^2).C_6$, $(C_5\times C_{15}^2):C_6$

Order 5400: $C_{15}^2.C_6.C_2^2$ x 3, $C_{15}^2.C_6.C_2^2$, $C_{15}^2:(C_4\times S_3)$

Order 4500: $C_5^3:(C_3^2:C_4)$ x 3, $C_3\times C_5^3:D_6$ x 2, $C_5^3:(C_3^2:C_4)$, $C_3^2\times D_5\times C_5:D_5$, $C_5^3:(C_3\times C_3:C_4)$, $C_3\times C_5^3:(C_3:C_4)$, $C_3\times C_5^3:D_6$, $(C_5^2\times C_{15}):C_{12}$, $C_5^2:C_{45}:C_4$, $C_5^3:C_3^2:C_4$

Order 4050: $(C_3\times C_{15}^2):S_3$ x 2, $(C_3\times C_{15}^2):C_6$

Order 3375: $C_{15}^3$, $C_{15}^2.C_{15}$, $C_{15}^2:C_{15}$

Order 3240: $C_3^3:(S_3\times F_5)$

Order 3000: $C_5^3:(C_2\times C_3:C_4)$ x 2, $C_5^3:(C_4\times S_3)$, $C_5^3:(C_4\times S_3)$

Order 2700: $C_5^2:(C_3^3:C_4)$ x 5, $C_5^2:(C_3\times C_3^2:C_4)$ x 3, $C_3^2\times D_{15}:D_5$ x 3, $C_5^2:(C_3\times C_3^2:C_4)$, $C_3^2\times S_3\times C_5:D_5$, $C_{15}^2:C_{12}$, $C_{15}^2:D_6$, $C_5^2:C_9:C_{12}$, $C_{15}^2:C_3:C_4$

Order 2250: $C_3^2\times C_5^2:D_5$ x 3, $C_5\times C_{15}\times D_{15}$ x 2, $C_3\times C_5^3:S_3$ x 2, $C_3^2\times C_5\times C_5:D_5$, $C_{15}^2\times D_5$, $C_3\times C_5\wr S_3$, $C_3\times C_5^2:D_{15}$, $C_5^3:C_{18}$, $C_3\times C_5^3:C_6$

Order 2025: $C_{15}^2.C_3^2$

Order 1800: $(C_{15}\times D_{15}):C_4$ x 6, $C_{15}^2:(C_2\times C_4)$ x 3, $S_3\times C_{15}:F_5$ x 2, $C_2\times C_{15}^2:C_4$, $C_{15}:F_5:S_3$

Order 1620: $C_3\wr C_3:F_5$, $(C_3^2\times C_{15}):C_{12}$, $C_3\wr S_3\times D_5$

Order 1500: $(C_5\times C_{15}):F_5$ x 3, $C_{15}:D_5^2$ x 2, $(C_5\times C_{15}):F_5$ x 2, $C_{15}:D_5^2$, $C_5^3:C_{12}$, $C_5^3:D_6$, $C_5^3:C_{12}$, $C_5\wr C_3:C_4$

Order 1350: $C_{15}^2:C_6$ x 6, $C_{15}^2:C_6$ x 5, $C_{15}^2:S_3$ x 2, $C_{15}^2:C_6$, $C_{15}^2.C_6$, $C_{15}^2:C_6$, $S_3\times C_{15}^2$

Order 1125: $C_5\times C_{15}^2$ x 3, $C_5^3:C_3^2$, $C_5^2:C_{45}$

Order 1080: $S_3\times C_3^2:F_5$ x 4, $C_3^2:(S_3\times F_5)$

Order 1000: $C_5^3:(C_2\times C_4)$

Order 900: $C_{15}^2:C_4$ x 24, $C_{15}^2:C_2^2$ x 6, $C_{15}^2:C_4$ x 6, $C_3^2\times D_5^2$ x 3, $C_{15}^2:C_4$ x 3, $C_{15}^2:C_2^2$ x 2, $C_{15}^2:C_4$ x 2, $C_3\times C_5^2:D_6$, $(C_5\times C_{15}):C_{12}$, $C_5^2:C_3^2:C_4$, $C_5^2:C_9:C_4$, $C_5^2:C_6^2$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 750: $C_5^3:C_6$ x 3, $C_5^2:C_{30}$ x 2, $C_5^2:C_{30}$ x 2, $C_5^2:D_{15}$, $C_5^2\times D_{15}$, $C_5^3:C_6$, $C_5^2:D_{15}$, $C_5\wr S_3$

Order 675: $C_3\times C_{15}^2$ x 5, $C_{15}^2:C_3$, $C_{15}^2.C_3$

Order 648: $C_3^3:(C_4\times S_3)$

Order 600: $D_5^2.S_3$ x 6, $D_{15}:F_5$ x 3, $C_{30}:F_5$ x 2, $S_3\times C_5:F_5$, $C_5^2:(C_4\times S_3)$

Order 540: $C_3^3:F_5$ x 5, $C_{15}:C_6^2$ x 4, $C_3^3:F_5$ x 4, $\He_3:F_5$ x 2, $\He_3:D_{10}$, $C_{45}:C_{12}$

Order 500: $C_5:D_5^2$, $C_5^3:C_4$, $C_5^3:C_4$

Order 450: $C_{15}^2:C_2$ x 24, $C_{15}\times D_{15}$ x 12, $C_{15}:C_{30}$ x 6, $C_{15}:D_{15}$ x 2, $C_{15}:C_{30}$ x 2, $C_3\times C_5^2:S_3$ x 2, $C_{15}\times C_{30}$, $C_3\times C_5^2:C_6$, $C_5^2:C_{18}$

Order 405: $C_3^3:C_{15}$

Order 375: $C_5^2\times C_{15}$ x 3, $C_5\wr C_3$

Order 360: $D_5.S_3^2$ x 9, $C_2\times C_3^2:F_5$ x 4

Order 324: $C_3^3:D_6$, $C_3\wr C_3:C_4$, $C_3^3:C_{12}$

Order 300: $C_{15}:F_5$ x 25, $C_3\times D_5^2$ x 6, $C_{15}:F_5$ x 6, $C_{15}:D_{10}$ x 3, $C_{15}:F_5$ x 3, $C_{15}:D_{10}$ x 2, $C_{15}:D_{10}$, $C_{15}:F_5$, $C_5^2:D_6$, $C_5^2:C_{12}$, $C_5^2:C_3:C_4$

Order 270: $D_5\times C_3^3$ x 5, $C_3^2\times D_{15}$ x 4, $C_3^2:C_{30}$ x 4, $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 250: $C_5^3:C_2$, $C_5^2:C_{10}$, $D_5\times C_5^2$

Order 225: $C_{15}^2$ x 23, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $C_6.S_3^2$, $C_6.S_3^2$

Order 200: $D_5:F_5$ x 3, $C_{10}:F_5$

Order 180: $C_3^2:F_5$ x 28, $C_{15}:D_6$ x 9, $C_{15}:C_{12}$ x 9, $C_3^2\times D_{10}$ x 4, $C_{45}:C_4$

Order 162: $C_3\wr S_3$ x 2, $C_3^3:C_6$

Order 150: $C_{15}:D_5$ x 25, $D_5\times C_{15}$ x 12, $C_5\times D_{15}$ x 6, $C_5^2:S_3$ x 2, $C_5\times C_{30}$ x 2, $C_5^2:C_6$, $C_5:D_{15}$, $S_3\times C_5^2$

Order 135: $C_3^2\times C_{15}$ x 5, $C_9:C_{15}$, $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $C_{30}:C_4$ x 8, $S_3\times F_5$ x 5

Order 108: $C_3^2:C_{12}$ x 2, $C_9:C_{12}$, $C_3^2\times D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:D_6$

Order 100: $C_5:F_5$ x 6, $D_5^2$ x 3, $C_5:F_5$ x 3, $C_5:D_{10}$

Order 90: $C_3^2\times D_5$ x 28, $C_3\times D_{15}$ x 9, $S_3\times C_{15}$ x 9, $C_3\times C_{30}$ x 4, $C_9\times D_5$

Order 81: $C_3\wr C_3$

Order 75: $C_5\times C_{15}$ x 23, $C_5^2:C_3$

Order 72: $C_6.D_6$ x 3, $C_6.D_6$

Order 60: $C_{15}:C_4$ x 32, $C_3\times D_{10}$ x 8, $S_3\times D_5$ x 5, $C_3\times F_5$ x 5

Order 54: $C_3^2:S_3$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 50: $C_5:D_5$ x 6, $C_5\times D_5$ x 6, $C_5\times C_{10}$

Order 45: $C_3\times C_{15}$ x 24, $C_{45}$

Order 40: $C_2\times F_5$ x 4

Order 36: $C_3^2:C_4$ x 5, $C_3:C_{12}$ x 3, $C_6\times S_3$ x 3, $C_6^2$, $C_9:C_4$

Order 30: $C_3\times D_5$ x 32, $C_{30}$ x 8, $D_{15}$ x 5, $C_5\times S_3$ x 5

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 25: $C_5^2$ x 5

Order 24: $C_6:C_4$ x 2, $C_4\times S_3$ x 2

Order 20: $F_5$ x 9, $D_{10}$ x 4

Order 18: $C_3\times C_6$ x 6, $C_3\times S_3$ x 6, $C_{18}$

Order 15: $C_{15}$ x 24

Order 12: $C_3:C_4$ x 6, $C_2\times C_6$ x 2, $D_6$ x 2, $C_{12}$ x 2

Order 10: $D_5$ x 9, $C_{10}$ x 4

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 8, $S_3$ x 4

Order 5: $C_5$ x 5

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 81000: $C_{15}\wr S_3:C_4$ ($C_1$)

Order 40500: $C_{15}\wr S_3:C_2$ ($C_2$), $C_{15}^3.C_{12}$ ($C_2$), $C_{15}\wr C_3:C_4$ ($C_2$)

Order 20250: $C_{15}\wr S_3$ ($C_4$), $(C_3\times C_{15}^2):D_{15}$ ($C_4$), $C_{15}\wr C_3:C_2$ ($C_2^2$)

Order 13500: $C_{15}^3.C_4$ ($S_3$), $(C_5\times C_{15}^2):D_6$ ($S_3$)

Order 10125: $C_{15}\wr C_3$ ($C_2\times C_4$)

Order 6750: $C_3^3\times C_5^2:D_5$ ($D_6$), $C_5\times C_{15}^2:S_3$ ($C_3:C_4$), $C_{15}^2:D_{15}$ ($C_3:C_4$), $(C_5\times C_{15}^2):C_6$ ($D_6$)

Order 4050: $(C_3\times C_{15}^2):S_3$ ($F_5$)

Order 3375: $C_{15}^3$ ($C_4\times S_3$), $C_{15}^2:C_{15}$ ($C_6:C_4$)

Order 2250: $C_3^2\times C_5^2:D_5$ ($S_3^2$)

Order 2025: $C_{15}^2.C_3^2$ ($C_2\times F_5$)

Order 1350: $C_{15}^2:S_3$ ($C_{15}:C_4$)

Order 1125: $C_5\times C_{15}^2$ ($C_6.D_6$)

Order 750: $C_5^3:C_6$ ($C_3^2:D_6$)

Order 675: $C_3\times C_{15}^2$ ($S_3\times F_5$), $C_{15}^2:C_3$ ($C_{30}:C_4$)

Order 375: $C_5^2\times C_{15}$ ($C_6.S_3^2$)

Order 250: $C_5^3:C_2$ ($C_3^3:D_6$)

Order 225: $C_{15}^2$ ($D_5.S_3^2$)

Order 135: $C_3^2\times C_{15}$ ($C_5^2:(C_4\times S_3)$)

Order 125: $C_5^3$ ($C_3^3:(C_4\times S_3)$)

Order 75: $C_5\times C_{15}$ ($C_3^2:(S_3\times F_5)$)

Order 45: $C_3\times C_{15}$ ($C_{15}:F_5:S_3$)

Order 27: $C_3^3$ ($C_5^3:(C_4\times S_3)$)

Order 25: $C_5^2$ ($C_3^3:(S_3\times F_5)$)

Order 15: $C_{15}$ ($C_{15}^2:(C_4\times S_3)$)

Order 9: $C_3^2$ ($C_5^3:C_6.D_6$)

Order 5: $C_5$ ($C_{15}^2:C_6.D_6$)

Order 3: $C_3$ ($C_{15}^2:(S_3\times F_5)$)

Order 1: $C_1$ ($C_{15}\wr S_3:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 81000: $C_{15}\wr S_3:C_4$ ($C_1$)

Order 40500: $C_{15}\wr S_3:C_2$ ($C_2$), $C_{15}^3.C_{12}$ ($C_2$), $C_{15}\wr C_3:C_4$ ($C_2$)

Order 20250: $C_{15}\wr S_3$ ($C_4$), $(C_3\times C_{15}^2):D_{15}$ ($C_4$), $C_{15}\wr C_3:C_2$ ($C_2^2$)

Order 13500: $C_{15}^3.C_4$ ($S_3$), $(C_5\times C_{15}^2):D_6$ ($S_3$)

Order 10125: $C_{15}\wr C_3$ ($C_2\times C_4$)

Order 6750: $C_3^3\times C_5^2:D_5$ ($D_6$), $C_5\times C_{15}^2:S_3$ ($C_3:C_4$), $C_{15}^2:D_{15}$ ($C_3:C_4$), $(C_5\times C_{15}^2):C_6$ ($D_6$)

Order 4050: $(C_3\times C_{15}^2):S_3$ ($F_5$)

Order 3375: $C_{15}^3$ ($C_4\times S_3$), $C_{15}^2:C_{15}$ ($C_6:C_4$)

Order 2250: $C_3^2\times C_5^2:D_5$ ($S_3^2$)

Order 2025: $C_{15}^2.C_3^2$ ($C_2\times F_5$)

Order 1350: $C_{15}^2:S_3$ ($C_{15}:C_4$)

Order 1125: $C_5\times C_{15}^2$ ($C_6.D_6$)

Order 750: $C_5^3:C_6$ ($C_3^2:D_6$)

Order 675: $C_3\times C_{15}^2$ ($S_3\times F_5$), $C_{15}^2:C_3$ ($C_{30}:C_4$)

Order 375: $C_5^2\times C_{15}$ ($C_6.S_3^2$)

Order 250: $C_5^3:C_2$ ($C_3^3:D_6$)

Order 225: $C_{15}^2$ ($D_5.S_3^2$)

Order 135: $C_3^2\times C_{15}$ ($C_5^2:(C_4\times S_3)$)

Order 125: $C_5^3$ ($C_3^3:(C_4\times S_3)$)

Order 75: $C_5\times C_{15}$ ($C_3^2:(S_3\times F_5)$)

Order 45: $C_3\times C_{15}$ ($C_{15}:F_5:S_3$)

Order 27: $C_3^3$ ($C_5^3:(C_4\times S_3)$)

Order 25: $C_5^2$ ($C_3^3:(S_3\times F_5)$)

Order 15: $C_{15}$ ($C_{15}^2:(C_4\times S_3)$)

Order 9: $C_3^2$ ($C_5^3:C_6.D_6$)

Order 5: $C_5$ ($C_{15}^2:C_6.D_6$)

Order 3: $C_3$ ($C_{15}^2:(S_3\times F_5)$)

Order 1: $C_1$ ($C_{15}\wr S_3:C_4$)

Series

Derived series

$C_{15}\wr S_3:C_4$

$\rhd$

$C_{15}\wr S_3:C_4$

$\rhd$

$C_{15}\wr C_3$

$\rhd$

$C_{15}\wr C_3$

$\rhd$

$C_{15}^2$

$\rhd$

$C_{15}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{15}\wr S_3:C_4$

$\rhd$

$C_{15}\wr S_3:C_4$

$\rhd$

$C_{15}\wr S_3:C_2$

$\rhd$

$C_{15}\wr S_3:C_2$

$\rhd$

$C_{15}\wr C_3:C_2$

$\rhd$

$C_{15}\wr C_3:C_2$

$\rhd$

$C_{15}\wr C_3$

$\rhd$

$C_{15}\wr C_3$

$\rhd$

$C_{15}^2:C_{15}$

$\rhd$

$C_{15}^2:C_{15}$

$\rhd$

$C_5\times C_{15}^2$

$\rhd$

$C_5\times C_{15}^2$

$\rhd$

$C_{15}^2$

$\rhd$

$C_{15}^2$

$\rhd$

$C_5\times C_{15}$

$\rhd$

$C_5\times C_{15}$

$\rhd$

$C_5^2$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{15}\wr S_3:C_4$

$\rhd$

$C_{15}\wr S_3:C_4$

$\rhd$

$C_{15}\wr C_3$

$\rhd$

$C_{15}\wr C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $272 \times 272$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $161 \times 161$ rational character table (warning: may be slow to load).