Abstract group 162000.x: $(C_{15}^3.A_4):C_4$

• Label: 162000.x
• Order: \( 2^{4} \cdot 3^{4} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $24$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$(C_{15}^3.A_4):C_4$
Order:	\(162000\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4, $C_5$ x 3
Derived length:	$4$

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	20	30	45
Elements	1	935	1826	12600	124	13870	3600	5940	36000	10424	18000	32400	11880	14400	162000
Conjugacy classes	1	3	5	4	6	7	2	6	10	74	2	4	7	4	135
Divisions	1	3	4	2	5	6	1	4	3	32	1	1	4	1	68
Autjugacy classes	1	3	4	4	5	6	1	4	6	32	1	2	4	1	74

Dimension	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Irr. complex chars.	4	2	4	8	8	4	16	9	16	0	64	0	0	0	135
Irr. rational chars.	2	3	2	0	3	2	7	4	8	2	11	1	16	7	68

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$14508$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	24	24
Arbitrary	16	18	18

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=c^{2}=d^{30}=e^{15}=f^{15}=[c,e]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(2,3)(4,6)(5,7,10,13)(8,11)(9,12)(14,15)(17,18,20,19)(21,22)(23,24), (1,2,4)(3,5,8)(6,9,7)(10,14,15)(11,13,12)(16,17,19,21,23,24,22,20,18)\rangle$
Transitive group:	45T1152				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{15}^3.A_4)$ $\,\rtimes\,$ $C_4$	$C_3^3$ $\,\rtimes\,$ $(D_5^3.S_3)$	$C_5^3$ $\,\rtimes\,$ $((C_3^2\times C_6).S_4)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{15}^3.C_2^3)$ . $S_3$	$(C_3^3\times C_5^2:D_5)$ . $S_4$	$(C_3^3:D_5\wr C_3)$ . $C_2$	$(C_5^3:C_2)$ . $(C_3^3:S_4)$	more information

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

Homology

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

Subgroups

There are 310712 subgroups in 980 conjugacy classes, 11 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_{15}^3.A_4):C_4$
Commutator:	$G' \simeq$ $C_{15}^3.A_4$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $(C_{15}^3.A_4):C_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_{15}^3$	$G/\operatorname{Fit} \simeq$ $A_4:C_4$
Radical:	$R \simeq$ $(C_{15}^3.A_4):C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{15}^3$	$G/\operatorname{soc} \simeq$ $A_4:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_4$
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

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 162000: $(C_{15}^3.A_4):C_4$

Order 81000: $C_3^3:D_5\wr C_3$

Order 54000: $C_{15}^3.C_2.D_4$

Order 40500: $C_{15}^3.A_4$, $C_{15}\wr C_3:C_4$

Order 27000: $C_{15}^3.C_4.C_2$, $C_{15}^3.C_2^3$, $C_3\times C_5^3.(C_2.S_3^2)$

Order 20250: $C_{15}\wr C_3:C_2$

Order 18000: $C_5^3.S_3^2:C_4$

Order 13500: $(C_3\times C_{15}^2).C_{20}$, $C_{15}^3.C_2^2$, $C_3\times C_5^3:(C_2\times C_3:S_3)$, $C_3\times C_5^3:(C_3\times C_3:C_4)$, $C_{15}^3.C_2^2$, $C_{15}^2:C_{15}:C_4$

Order 10800: $(C_3\times C_{15}^2).C_2.D_4$

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

Order 9000: $C_3\times C_5^3:(C_4\times S_3)$, $C_5^3.C_3:S_3.C_2^2$, $C_5^3.(C_2.S_3^2)$, $C_5^3.(C_2\times S_3^2)$

Order 8100: $C_{15}^2:C_3:C_{12}$

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

Order 6000: $C_5^3.C_6.D_4$, $D_5^3.S_3$

Order 5400: $(C_3\times C_{15}^2).C_4.C_2$, $C_{15}^2.C_6.C_2^2$, $(C_3\times C_{15}^2).C_4.C_2$, $C_3\times C_5^2:(C_2.S_3^2)$, $C_3\times C_5^2:(C_2.S_3^2)$

Order 4500: $C_5^3:(C_3\times C_3:C_4)$ x 2, $C_3\times C_5^3:D_6$ x 2, $C_5^3:S_3^2$, $C_5^3:S_3^2$, $C_3\times C_5^3:(C_3:C_4)$, $C_5^3:(C_3:S_3.C_2)$, $C_5^3:(C_2\times C_3:S_3)$, $C_3^2\times C_5:(C_5:F_5)$, $C_5^3:S_3^2$, $C_5\wr C_3:C_{12}$

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

Order 3600: $D_{15}^2:C_4$

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

Order 3000: $C_5^3:(C_2\times C_3:C_4)$, $C_5^3:(C_2^2\times S_3)$, $C_5^3:(C_4\times S_3)$, $C_3\times C_5^3:(C_2\times C_4)$, $D_5\wr C_3$

Order 2700: $C_3\times C_5^2:(C_3\times C_3:C_4)$ x 4, $C_3\times C_5^2:(C_2\times C_3:S_3)$ x 3, $C_5^2:(C_3^3:C_2^2)$ x 2, $C_3\times C_5^2:(C_3\times C_3:C_4)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5\times C_5:(C_3^2:(C_3:C_4))$, $C_3\times C_5^2:(C_2\times C_3:S_3)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_3^3:C_2^2)$, $C_{15}^2:C_3:C_4$

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, $C_5^2\times C_3:D_{15}$, $C_5^3:(C_3:S_3)$, $C_5^3:C_{18}$, $C_3\times C_5^3:C_6$

Order 2160: $(D_5\times C_3^3).D_4$

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

Order 2000: $D_5^2:F_5$

Order 1800: $D_{15}^2:C_2$, $C_{15}^2:C_2^3$, $C_{15}^2:(C_2\times C_4)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_3\times D_{15}:F_5$, $C_3\times D_{15}:F_5$, $S_3\times C_{15}:F_5$

Order 1620: $(C_5\times \He_3):C_{12}$

Order 1500: $C_5^3:C_{12}$ x 3, $C_{15}:D_5^2$ x 2, $C_5^2:D_{30}$, $C_{15}:D_5^2$, $C_5^2:D_{30}$, $C_{15}:D_5^2$, $(C_5\times C_{15}):F_5$, $C_5\times C_{15}:F_5$, $C_5^3:A_4$, $C_5\wr C_3:C_4$

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

Order 1296: $(C_3^2\times C_6).S_4$

Order 1200: $(C_5\times C_{10}).D_{12}$

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

Order 1080: $C_{15}:(S_3\times C_{12})$ x 2, $D_5\times C_3:S_3^2$, $C_3^3:C_4\times D_5$, $(D_5\times C_3^3):C_4$

Order 1000: $D_5^3$, $D_5.D_5^2$, $C_5^3:(C_2\times C_4)$

Order 900: $C_{15}^2:C_4$ x 14, $C_{15}^2:C_2^2$ x 8, $C_{15}^2:C_4$ x 4, $C_{15}^2:C_2^2$ x 3, $C_{15}^2:C_4$ x 3, $C_{15}:D_{30}$ x 2, $D_{15}^2$ x 2, $C_{15}:D_{30}$ x 2, $C_{15}^2:C_2^2$ x 2, $C_5^2:C_3:C_{12}$, $C_5^2:S_3^2$, $C_{15}^2:C_2^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$

Order 810: $C_3\wr C_3\times D_5$

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

Order 720: $S_3^2:F_5$

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

Order 648: $C_2\times C_3^3:A_4$

Order 600: $C_{30}:D_{10}$, $C_{30}:F_5$, $S_3\times D_5^2$, $D_5^2.S_3$, $D_{15}:F_5$, $D_{15}:F_5$, $S_3\times C_5:F_5$, $D_5^2.C_6$, $C_{30}.D_{10}$

Order 540: $C_3^3:F_5$ x 5, $C_3^3:D_{10}$ x 4, $C_{15}:S_3^2$ x 2, $(C_3^2\times C_{15}):C_4$, $C_3^3:F_5$, $C_3^3:C_{20}$, $\He_3:F_5$, $C_{15}:S_3^2$

Order 500: $C_5:D_5^2$, $C_5:D_5^2$, $C_5\times D_5^2$, $C_5^3:C_4$, $C_5^2:C_{20}$

Order 450: $C_{15}^2:C_2$ x 31, $C_{15}\times D_{15}$ x 16, $C_{15}:D_{15}$ x 6, $C_{15}:D_{15}$ x 2, $C_{15}:C_{30}$ x 2, $C_{15}:D_{15}$, $C_{15}^2:C_2$, $C_3\times C_5^2:C_6$, $C_5^2:C_{18}$

Order 432: $C_6.\SOPlus(4,2)$

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

Order 400: $D_5^2:C_4$

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

Order 360: $D_{15}:C_{12}$ x 3, $D_5\times S_3^2$ x 2, $D_5.S_3^2$ x 2, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$

Order 324: $C_3^3:A_4$, $\He_3:C_{12}$

Order 300: $C_{15}:F_5$ x 14, $C_{15}:D_{10}$ x 10, $C_{15}:F_5$ x 4, $C_3\times D_5^2$ x 3, $C_{15}:F_5$ x 3, $D_5\times D_{15}$ x 2, $C_{15}:D_{10}$ x 2, $C_{15}:D_{10}$ x 2, $C_5\times D_{30}$, $C_{15}:D_{10}$, $C_{15}:F_5$, $C_{15}:F_5$, $C_{15}:C_{20}$, $C_5^2:C_3:C_4$, $C_{15}:C_{20}$

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

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

Order 240: $D_{30}:C_4$

Order 225: $C_{15}^2$ x 31, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $C_6:S_3^2$, $C_2\times C_3^3:C_4$, $C_6.S_3^2$

Order 200: $D_5\times D_{10}$, $C_{10}:F_5$, $D_5:F_5$, $D_5\times F_5$, $C_{10}.D_{10}$

Order 180: $C_{15}:C_{12}$ x 18, $C_{15}:D_6$ x 10, $C_3^2\times F_5$ x 5, $C_{15}:D_6$ x 4, $S_3\times D_{15}$ x 3, $C_{15}:D_6$ x 2, $C_5\times S_3^2$ x 2, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^3:C_6$

Order 150: $C_{15}:D_5$ x 32, $C_5\times D_{15}$ x 16, $D_5\times C_{15}$ x 6, $C_5:D_{15}$ x 2, $S_3\times C_5^2$ x 2, $C_5^2:C_6$, $C_5\times C_{30}$

Order 144: $S_3^2:C_4$

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: $S_3\times F_5$ x 3, $S_3\times D_{10}$ x 2, $C_6\times F_5$ x 2, $C_6.D_{10}$, $C_{30}:C_4$

Order 108: $C_3:S_3^2$ x 2, $C_3^2:D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$, $\He_3:C_4$

Order 100: $D_5^2$ x 5, $C_5:F_5$ x 4, $C_5\times F_5$, $C_5:C_{20}$, $C_5:D_{10}$, $C_5\times D_{10}$, $C_5:F_5$, $C_5:F_5$

Order 90: $C_3^2\times D_5$ x 32, $C_3\times D_{15}$ x 10, $S_3\times C_{15}$ x 10, $C_3:D_{15}$ x 4, $C_{15}:S_3$ x 4, $C_9\times D_5$

Order 81: $C_3\wr C_3$

Order 80: $D_{10}:C_4$

Order 75: $C_5\times C_{15}$ x 31, $C_5^2:C_3$

Order 72: $S_3\times D_6$, $C_2\times C_3^2:C_4$, $S_3\times C_{12}$, $C_6.D_6$

Order 60: $C_3\times F_5$ x 17, $S_3\times D_5$ x 13, $C_{15}:C_4$ x 7, $C_3\times D_{10}$ x 4, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_{15}:C_4$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ 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 48: $A_4:C_4$, $D_6:C_4$

Order 45: $C_3\times C_{15}$ x 32, $C_{45}$

Order 40: $C_2\times F_5$ x 3, $C_4\times D_5$, $C_2\times D_{10}$

Order 36: $C_3:C_{12}$ x 4, $S_3^2$ x 3, $C_6\times S_3$ x 2, $C_3^2:C_4$, $C_3\times C_{12}$, $C_6:S_3$

Order 30: $C_3\times D_5$ x 36, $D_{15}$ x 10, $C_5\times S_3$ x 10, $C_{30}$ x 4

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

Order 25: $C_5^2$ x 5

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$, $C_2\times D_6$, $C_2\times A_4$

Order 20: $D_{10}$ x 6, $F_5$ x 6, $C_2\times C_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 4, $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_{18}$

Order 16: $C_2^2:C_4$

Order 15: $C_{15}$ x 32

Order 12: $D_6$ x 5, $C_{12}$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$, $A_4$

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

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

Order 8: $C_2\times C_4$ x 2, $C_2^3$

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

Order 5: $C_5$ x 5

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 162000: $(C_{15}^3.A_4):C_4$

Order 81000: $C_3^3:D_5\wr C_3$

Order 54000: $C_{15}^3.C_2.D_4$

Order 40500: $C_{15}^3.A_4$, $C_{15}\wr C_3:C_4$

Order 27000: $C_{15}^3.C_4.C_2$, $C_{15}^3.C_2^3$, $C_3\times C_5^3.(C_2.S_3^2)$

Order 20250: $C_{15}\wr C_3:C_2$

Order 18000: $C_5^3.S_3^2:C_4$

Order 13500: $(C_3\times C_{15}^2).C_{20}$, $C_{15}^3.C_2^2$, $C_3\times C_5^3:(C_2\times C_3:S_3)$, $C_3\times C_5^3:(C_3\times C_3:C_4)$, $C_{15}^3.C_2^2$, $C_{15}^2:C_{15}:C_4$

Order 10800: $(C_3\times C_{15}^2).C_2.D_4$

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

Order 9000: $C_3\times C_5^3:(C_4\times S_3)$, $C_5^3.C_3:S_3.C_2^2$, $C_5^3.(C_2.S_3^2)$, $C_5^3.(C_2\times S_3^2)$

Order 8100: $C_{15}^2:C_3:C_{12}$

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

Order 6000: $C_5^3.C_6.D_4$, $D_5^3.S_3$

Order 5400: $(C_3\times C_{15}^2).C_4.C_2$, $C_{15}^2.C_6.C_2^2$, $(C_3\times C_{15}^2).C_4.C_2$, $C_3\times C_5^2:(C_2.S_3^2)$, $C_3\times C_5^2:(C_2.S_3^2)$

Order 4500: $C_5^3:(C_3\times C_3:C_4)$ x 2, $C_3\times C_5^3:D_6$ x 2, $C_5^3:S_3^2$, $C_5^3:S_3^2$, $C_3\times C_5^3:(C_3:C_4)$, $C_5^3:(C_3:S_3.C_2)$, $C_5^3:(C_2\times C_3:S_3)$, $C_3^2\times C_5:(C_5:F_5)$, $C_5^3:S_3^2$, $C_5\wr C_3:C_{12}$

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

Order 3600: $D_{15}^2:C_4$

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

Order 3000: $C_5^3:(C_2\times C_3:C_4)$, $C_5^3:(C_2^2\times S_3)$, $C_5^3:(C_4\times S_3)$, $C_3\times C_5^3:(C_2\times C_4)$, $D_5\wr C_3$

Order 2700: $C_3\times C_5^2:(C_3\times C_3:C_4)$ x 4, $C_3\times C_5^2:(C_2\times C_3:S_3)$ x 3, $C_5^2:(C_3^3:C_2^2)$ x 2, $C_3\times C_5^2:(C_3\times C_3:C_4)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5\times C_5:(C_3^2:(C_3:C_4))$, $C_3\times C_5^2:(C_2\times C_3:S_3)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_3^3:C_2^2)$, $C_{15}^2:C_3:C_4$

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, $C_5^2\times C_3:D_{15}$, $C_5^3:(C_3:S_3)$, $C_5^3:C_{18}$, $C_3\times C_5^3:C_6$

Order 2160: $(D_5\times C_3^3).D_4$

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

Order 2000: $D_5^2:F_5$

Order 1800: $D_{15}^2:C_2$, $C_{15}^2:C_2^3$, $C_{15}^2:(C_2\times C_4)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_3\times D_{15}:F_5$, $C_3\times D_{15}:F_5$, $S_3\times C_{15}:F_5$

Order 1620: $(C_5\times \He_3):C_{12}$

Order 1500: $C_5^3:C_{12}$ x 3, $C_{15}:D_5^2$ x 2, $C_5^2:D_{30}$, $C_{15}:D_5^2$, $C_5^2:D_{30}$, $C_{15}:D_5^2$, $(C_5\times C_{15}):F_5$, $C_5\times C_{15}:F_5$, $C_5^3:A_4$, $C_5\wr C_3:C_4$

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

Order 1296: $(C_3^2\times C_6).S_4$

Order 1200: $(C_5\times C_{10}).D_{12}$

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

Order 1080: $C_{15}:(S_3\times C_{12})$ x 2, $D_5\times C_3:S_3^2$, $C_3^3:C_4\times D_5$, $(D_5\times C_3^3):C_4$

Order 1000: $D_5^3$, $D_5.D_5^2$, $C_5^3:(C_2\times C_4)$

Order 900: $C_{15}^2:C_4$ x 14, $C_{15}^2:C_2^2$ x 8, $C_{15}^2:C_4$ x 4, $C_{15}^2:C_2^2$ x 3, $C_{15}^2:C_4$ x 3, $C_{15}:D_{30}$ x 2, $D_{15}^2$ x 2, $C_{15}:D_{30}$ x 2, $C_{15}^2:C_2^2$ x 2, $C_5^2:C_3:C_{12}$, $C_5^2:S_3^2$, $C_{15}^2:C_2^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$

Order 810: $C_3\wr C_3\times D_5$

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

Order 720: $S_3^2:F_5$

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

Order 648: $C_2\times C_3^3:A_4$

Order 600: $C_{30}:D_{10}$, $C_{30}:F_5$, $S_3\times D_5^2$, $D_5^2.S_3$, $D_{15}:F_5$, $D_{15}:F_5$, $S_3\times C_5:F_5$, $D_5^2.C_6$, $C_{30}.D_{10}$

Order 540: $C_3^3:F_5$ x 5, $C_3^3:D_{10}$ x 4, $C_{15}:S_3^2$ x 2, $(C_3^2\times C_{15}):C_4$, $C_3^3:F_5$, $C_3^3:C_{20}$, $\He_3:F_5$, $C_{15}:S_3^2$

Order 500: $C_5:D_5^2$, $C_5:D_5^2$, $C_5\times D_5^2$, $C_5^3:C_4$, $C_5^2:C_{20}$

Order 450: $C_{15}^2:C_2$ x 31, $C_{15}\times D_{15}$ x 16, $C_{15}:D_{15}$ x 6, $C_{15}:D_{15}$ x 2, $C_{15}:C_{30}$ x 2, $C_{15}:D_{15}$, $C_{15}^2:C_2$, $C_3\times C_5^2:C_6$, $C_5^2:C_{18}$

Order 432: $C_6.\SOPlus(4,2)$

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

Order 400: $D_5^2:C_4$

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

Order 360: $D_{15}:C_{12}$ x 3, $D_5\times S_3^2$ x 2, $D_5.S_3^2$ x 2, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$

Order 324: $C_3^3:A_4$, $\He_3:C_{12}$

Order 300: $C_{15}:F_5$ x 14, $C_{15}:D_{10}$ x 10, $C_{15}:F_5$ x 4, $C_3\times D_5^2$ x 3, $C_{15}:F_5$ x 3, $D_5\times D_{15}$ x 2, $C_{15}:D_{10}$ x 2, $C_{15}:D_{10}$ x 2, $C_5\times D_{30}$, $C_{15}:D_{10}$, $C_{15}:F_5$, $C_{15}:F_5$, $C_{15}:C_{20}$, $C_5^2:C_3:C_4$, $C_{15}:C_{20}$

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

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

Order 240: $D_{30}:C_4$

Order 225: $C_{15}^2$ x 31, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $C_6:S_3^2$, $C_2\times C_3^3:C_4$, $C_6.S_3^2$

Order 200: $D_5\times D_{10}$, $C_{10}:F_5$, $D_5:F_5$, $D_5\times F_5$, $C_{10}.D_{10}$

Order 180: $C_{15}:C_{12}$ x 18, $C_{15}:D_6$ x 10, $C_3^2\times F_5$ x 5, $C_{15}:D_6$ x 4, $S_3\times D_{15}$ x 3, $C_{15}:D_6$ x 2, $C_5\times S_3^2$ x 2, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^3:C_6$

Order 150: $C_{15}:D_5$ x 32, $C_5\times D_{15}$ x 16, $D_5\times C_{15}$ x 6, $C_5:D_{15}$ x 2, $S_3\times C_5^2$ x 2, $C_5^2:C_6$, $C_5\times C_{30}$

Order 144: $S_3^2:C_4$

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: $S_3\times F_5$ x 3, $S_3\times D_{10}$ x 2, $C_6\times F_5$ x 2, $C_6.D_{10}$, $C_{30}:C_4$

Order 108: $C_3:S_3^2$ x 2, $C_3^2:D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$, $\He_3:C_4$

Order 100: $D_5^2$ x 5, $C_5:F_5$ x 4, $C_5\times F_5$, $C_5:C_{20}$, $C_5:D_{10}$, $C_5\times D_{10}$, $C_5:F_5$, $C_5:F_5$

Order 90: $C_3^2\times D_5$ x 32, $C_3\times D_{15}$ x 10, $S_3\times C_{15}$ x 10, $C_3:D_{15}$ x 4, $C_{15}:S_3$ x 4, $C_9\times D_5$

Order 81: $C_3\wr C_3$

Order 80: $D_{10}:C_4$

Order 75: $C_5\times C_{15}$ x 31, $C_5^2:C_3$

Order 72: $S_3\times D_6$, $C_2\times C_3^2:C_4$, $S_3\times C_{12}$, $C_6.D_6$

Order 60: $C_3\times F_5$ x 17, $S_3\times D_5$ x 13, $C_{15}:C_4$ x 7, $C_3\times D_{10}$ x 4, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_{15}:C_4$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ 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 48: $A_4:C_4$, $D_6:C_4$

Order 45: $C_3\times C_{15}$ x 32, $C_{45}$

Order 40: $C_2\times F_5$ x 3, $C_4\times D_5$, $C_2\times D_{10}$

Order 36: $C_3:C_{12}$ x 4, $S_3^2$ x 3, $C_6\times S_3$ x 2, $C_3^2:C_4$, $C_3\times C_{12}$, $C_6:S_3$

Order 30: $C_3\times D_5$ x 36, $D_{15}$ x 10, $C_5\times S_3$ x 10, $C_{30}$ x 4

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

Order 25: $C_5^2$ x 5

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$, $C_2\times D_6$, $C_2\times A_4$

Order 20: $D_{10}$ x 6, $F_5$ x 6, $C_2\times C_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 4, $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_{18}$

Order 16: $C_2^2:C_4$

Order 15: $C_{15}$ x 32

Order 12: $D_6$ x 5, $C_{12}$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$, $A_4$

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

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

Order 8: $C_2\times C_4$ x 2, $C_2^3$

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

Order 5: $C_5$ x 5

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 162000: $(C_{15}^3.A_4):C_4$ ($C_1$)

Order 81000: $C_3^3:D_5\wr C_3$ ($C_2$)

Order 40500: $C_{15}^3.A_4$ ($C_4$)

Order 27000: $C_{15}^3.C_2^3$ ($S_3$)

Order 13500: $C_{15}^3.C_2^2$ ($C_3:C_4$)

Order 6750: $C_3^3\times C_5^2:D_5$ ($S_4$)

Order 3375: $C_{15}^3$ ($A_4:C_4$)

Order 250: $C_5^3:C_2$ ($C_3^3:S_4$)

Order 125: $C_5^3$ ($(C_3^2\times C_6).S_4$)

Order 27: $C_3^3$ ($D_5^3.S_3$)

Order 1: $C_1$ ($(C_{15}^3.A_4):C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 162000: $(C_{15}^3.A_4):C_4$ ($C_1$)

Order 81000: $C_3^3:D_5\wr C_3$ ($C_2$)

Order 40500: $C_{15}^3.A_4$ ($C_4$)

Order 27000: $C_{15}^3.C_2^3$ ($S_3$)

Order 13500: $C_{15}^3.C_2^2$ ($C_3:C_4$)

Order 6750: $C_3^3\times C_5^2:D_5$ ($S_4$)

Order 3375: $C_{15}^3$ ($A_4:C_4$)

Order 250: $C_5^3:C_2$ ($C_3^3:S_4$)

Order 125: $C_5^3$ ($(C_3^2\times C_6).S_4$)

Order 27: $C_3^3$ ($D_5^3.S_3$)

Order 1: $C_1$ ($(C_{15}^3.A_4):C_4$)

Series

Derived series

$(C_{15}^3.A_4):C_4$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3$

$\rhd$

$C_1$

Chief series

$(C_{15}^3.A_4):C_4$

$\rhd$

$C_3^3:D_5\wr C_3$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

Lower central series

$(C_{15}^3.A_4):C_4$

$\rhd$

$C_{15}^3.A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 5 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 $135 \times 135$ 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 $68 \times 68$ rational character table.