Abstract group 27000.b: $C_{15}^2:(C_6\times F_5)$

• Label: 27000.b
• Order: \( 2^{3} \cdot 3^{3} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_{15}^2:(C_6\times F_5)$
Order:	\(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$, of order \(1296000\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{3} \)
Composition factors:	$C_2$ x 3, $C_3$ x 3, $C_5$ x 3
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), and metabelian.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	1259	458	2500	124	5950	1116	11000	2792	1800	27000
Conjugacy classes	1	3	6	4	11	10	11	16	46	2	110
Divisions	1	3	4	2	11	6	11	5	44	1	88
Autjugacy classes	1	3	4	4	3	6	3	10	8	1	43

Dimension	1	2	4	6	8	12	16	24
Irr. complex chars.	24	12	6	4	3	20	0	41	110
Irr. rational chars.	4	8	7	2	4	21	1	41	88

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d \mid a^{12}=b^{10}=c^{15}=d^{15}=[c,d]=1, b^{a}=b^{9}c^{4}d, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,3,7,10,2,5,11,12,6,8,13,15)(4,9,14)(16,18,21)(17,19,22)(20,23,24), (1,2,4,8,12,5,10,13,7,9,3,6)(11,14,15)(16,17,19)(18,20,23,24,21,22)\rangle$
Transitive group:	45T668				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{15}^2:C_6)$ $\,\rtimes\,$ $F_5$	$C_{15}^2$ $\,\rtimes\,$ $(C_6\times F_5)$	$(C_{15}^2:C_{30})$ $\,\rtimes\,$ $C_4$	$(C_3\times C_5^3:C_{12})$ $\,\rtimes\,$ $S_3$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times C_5^3:C_6)$ . $D_6$	$C_3$ . $(S_3\times C_5^3:C_{12})$	$(C_5^3:C_6)$ . $(C_6\times S_3)$	$C_{15}$ . $(S_3\times C_5^2:C_{12})$	all 12

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

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 120912 subgroups in 1208 conjugacy classes, 40 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{15}^2:(C_6\times F_5)$
Commutator:	$G' \simeq$ $C_5\times C_{15}^2$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_3\times C_5^3:C_{12}$
Fitting:	$\operatorname{Fit} \simeq$ $C_5\times C_{15}^2$	$G/\operatorname{Fit} \simeq$ $C_2\times C_{12}$
Radical:	$R \simeq$ $C_{15}^2:(C_6\times F_5)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^2\times C_{15}$	$G/\operatorname{soc} \simeq$ $S_3\times C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$
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

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 27000: $C_{15}^2:(C_6\times F_5)$

Order 13500: $(C_5\times C_{15}^2):C_{12}$, $(C_5\times C_{15}^2):C_{12}$, $C_{15}^2:(C_3\times D_{10})$

Order 9000: $C_5^2:D_5.C_2\times C_3:S_3$, $S_3\times C_5^3:C_{12}$

Order 6750: $(C_5\times C_{15}^2):C_6$, $(C_5\times C_{15}^2):C_6$, $C_{15}^2:C_{30}$

Order 5400: $C_{15}^2:(C_2\times C_{12})$

Order 4500: $C_3\times C_5^3:C_{12}$ x 2, $C_3^2\times C_5^2:D_5.C_2$, $C_5^3:(C_3^2:C_4)$, $C_3:S_3\times C_5^2:D_5$, $(C_5^2\times C_{15}):C_{12}$, $C_5^3:C_6\times S_3$

Order 3375: $C_{15}^2:C_{15}$

Order 3000: $C_5^2:D_5.C_2\times S_3$ x 2, $C_2\times C_5^3:C_{12}$

Order 2700: $C_{15}^2:C_{12}$, $C_{15}^2:C_{12}$, $C_{15}^2:(C_2\times C_6)$

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

Order 1800: $C_{15}:(S_3\times F_5)$ x 11, $S_3\times C_5^2:C_{12}$

Order 1500: $C_5^3:C_{12}$ x 3, $C_5^3:D_6$ x 2, $(C_5\times C_{15}):F_5$ x 2, $C_5^3:C_{12}$ x 2, $C_2\times C_5^3:C_6$

Order 1350: $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$

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

Order 1080: $C_3^2:C_6\times F_5$

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

Order 900: $C_{15}^2:C_2^2$ x 11, $C_{15}^2:C_4$ x 11, $C_{15}^2:C_4$ x 11, $C_3\times C_5^2:C_{12}$ x 2, $C_5^2:C_6\times S_3$, $(C_5\times C_{15}):C_{12}$

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

Order 675: $C_{15}^2:C_3$

Order 600: $S_3\times C_5:F_5$ x 42, $C_2\times C_5^2:C_{12}$

Order 540: $\He_3:D_{10}$, $\He_3:F_5$, $F_5\times \He_3$

Order 500: $C_5^3:C_4$ x 2, $C_5^2:D_{10}$

Order 450: $C_{15}:D_{15}$ x 11, $C_{15}^2:C_2$ x 11, $C_{15}^2:C_2$ x 11, $C_3\times C_5^2:C_6$ x 2, $(C_5\times C_{15}):C_6$, $C_5^2:C_3\times S_3$

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

Order 360: $C_3:S_3\times F_5$ x 11, $D_{15}:C_{12}$

Order 300: $C_{15}:D_{10}$ x 42, $C_{15}:F_5$ x 42, $C_{15}:F_5$ x 42, $C_5^2:C_{12}$ x 3, $C_2\times C_5^2:C_6$

Order 270: $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$

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

Order 225: $C_{15}^2$ x 11, $C_5^2:C_3^2$ x 2

Order 216: $C_4\times C_3^2:C_6$

Order 200: $C_{10}:F_5$ x 11

Order 180: $C_3^2\times F_5$ x 13, $C_{15}:D_6$ x 11, $C_3^2:F_5$ x 11, $C_{15}:D_6$, $C_{15}:C_{12}$

Order 150: $C_{15}:D_5$ x 42, $C_5:D_{15}$ x 42, $S_3\times C_5^2$ x 42, $C_5^2:C_6$ x 3, $C_5^2:C_6$

Order 135: $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $S_3\times F_5$ x 42, $C_6\times F_5$

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

Order 100: $C_5:F_5$ x 22, $C_5:D_{10}$ x 11

Order 90: $C_3^2\times D_5$ x 13, $C_3:D_{15}$ x 11, $C_{15}:S_3$ x 11, $C_3\times D_{15}$, $S_3\times C_{15}$

Order 75: $C_5\times C_{15}$ x 42, $C_5^2:C_3$ x 2

Order 72: $C_{12}:S_3$, $S_3\times C_{12}$

Order 60: $C_3\times F_5$ x 45, $S_3\times D_5$ x 42, $C_{15}:C_4$ x 42, $C_3\times D_{10}$

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

Order 50: $C_5:D_5$ x 22, $C_5\times C_{10}$ x 11

Order 45: $C_3\times C_{15}$ x 13

Order 40: $C_2\times F_5$ x 11

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

Order 30: $C_3\times D_5$ x 45, $D_{15}$ x 42, $C_5\times S_3$ x 42, $C_{30}$

Order 27: $\He_3$

Order 25: $C_5^2$ x 11

Order 24: $C_4\times S_3$ x 2, $C_2\times C_{12}$

Order 20: $F_5$ x 22, $D_{10}$ x 11

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 2, $C_3\times S_3$ x 2

Order 15: $C_{15}$ x 44

Order 12: $C_{12}$ x 5, $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 10: $D_5$ x 22, $C_{10}$ x 11

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$

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

Order 5: $C_5$ x 11

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 27000: $C_{15}^2:(C_6\times F_5)$

Order 13500: $(C_5\times C_{15}^2):C_{12}$, $(C_5\times C_{15}^2):C_{12}$, $C_{15}^2:(C_3\times D_{10})$

Order 9000: $C_5^2:D_5.C_2\times C_3:S_3$, $S_3\times C_5^3:C_{12}$

Order 6750: $(C_5\times C_{15}^2):C_6$, $(C_5\times C_{15}^2):C_6$, $C_{15}^2:C_{30}$

Order 5400: $C_{15}^2:(C_2\times C_{12})$

Order 4500: $C_3\times C_5^3:C_{12}$ x 2, $C_3^2\times C_5^2:D_5.C_2$, $C_5^3:(C_3^2:C_4)$, $C_3:S_3\times C_5^2:D_5$, $(C_5^2\times C_{15}):C_{12}$, $C_5^3:C_6\times S_3$

Order 3375: $C_{15}^2:C_{15}$

Order 3000: $C_5^2:D_5.C_2\times S_3$ x 2, $C_2\times C_5^3:C_{12}$

Order 2700: $C_{15}^2:C_{12}$, $C_{15}^2:C_{12}$, $C_{15}^2:(C_2\times C_6)$

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

Order 1800: $C_{15}:(S_3\times F_5)$ x 3, $S_3\times C_5^2:C_{12}$

Order 1500: $C_5^3:C_{12}$ x 3, $C_5^3:D_6$ x 2, $(C_5\times C_{15}):F_5$ x 2, $C_5^3:C_{12}$ x 2, $C_2\times C_5^3:C_6$

Order 1350: $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$

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

Order 1080: $C_3^2:C_6\times F_5$

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

Order 900: $C_{15}^2:C_2^2$ x 3, $C_{15}^2:C_4$ x 3, $C_{15}^2:C_4$ x 3, $C_3\times C_5^2:C_{12}$ x 2, $C_5^2:C_6\times S_3$, $(C_5\times C_{15}):C_{12}$

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

Order 675: $C_{15}^2:C_3$

Order 600: $S_3\times C_5:F_5$ x 6, $C_2\times C_5^2:C_{12}$

Order 540: $\He_3:D_{10}$, $\He_3:F_5$, $F_5\times \He_3$

Order 500: $C_5^3:C_4$ x 2, $C_5^2:D_{10}$

Order 450: $C_{15}:D_{15}$ x 3, $C_{15}^2:C_2$ x 3, $C_{15}^2:C_2$ x 3, $C_3\times C_5^2:C_6$ x 2, $(C_5\times C_{15}):C_6$, $C_5^2:C_3\times S_3$

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

Order 360: $C_3:S_3\times F_5$ x 3, $D_{15}:C_{12}$

Order 300: $C_{15}:D_{10}$ x 6, $C_{15}:F_5$ x 6, $C_{15}:F_5$ x 6, $C_5^2:C_{12}$ x 3, $C_2\times C_5^2:C_6$

Order 270: $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$

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

Order 225: $C_{15}^2$ x 3, $C_5^2:C_3^2$ x 2

Order 216: $C_4\times C_3^2:C_6$

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

Order 180: $C_3^2\times F_5$ x 5, $C_{15}:D_6$ x 3, $C_3^2:F_5$ x 3, $C_{15}:D_6$, $C_{15}:C_{12}$

Order 150: $C_{15}:D_5$ x 6, $C_5:D_{15}$ x 6, $S_3\times C_5^2$ x 6, $C_5^2:C_6$ x 3, $C_5^2:C_6$

Order 135: $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $S_3\times F_5$ x 6, $C_6\times F_5$

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

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

Order 90: $C_3^2\times D_5$ x 5, $C_3:D_{15}$ x 3, $C_{15}:S_3$ x 3, $C_3\times D_{15}$, $S_3\times C_{15}$

Order 75: $C_5\times C_{15}$ x 6, $C_5^2:C_3$ x 2

Order 72: $C_{12}:S_3$, $S_3\times C_{12}$

Order 60: $C_3\times F_5$ x 9, $S_3\times D_5$ x 6, $C_{15}:C_4$ x 6, $C_3\times D_{10}$

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

Order 50: $C_5:D_5$ x 6, $C_5\times C_{10}$ x 3

Order 45: $C_3\times C_{15}$ x 5

Order 40: $C_2\times F_5$ x 3

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

Order 30: $C_3\times D_5$ x 9, $D_{15}$ x 6, $C_5\times S_3$ x 6, $C_{30}$

Order 27: $\He_3$

Order 25: $C_5^2$ x 3

Order 24: $C_4\times S_3$ x 2, $C_2\times C_{12}$

Order 20: $F_5$ x 6, $D_{10}$ x 3

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 2, $C_3\times S_3$ x 2

Order 15: $C_{15}$ x 8

Order 12: $C_{12}$ x 5, $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 10: $D_5$ x 6, $C_{10}$ x 3

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$

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

Order 5: $C_5$ x 3

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 27000: $C_{15}^2:(C_6\times F_5)$ ($C_1$)

Order 13500: $(C_5\times C_{15}^2):C_{12}$ ($C_2$), $(C_5\times C_{15}^2):C_{12}$ ($C_2$), $C_{15}^2:(C_3\times D_{10})$ ($C_2$)

Order 9000: $C_5^2:D_5.C_2\times C_3:S_3$ ($C_3$)

Order 6750: $(C_5\times C_{15}^2):C_6$ ($C_4$), $(C_5\times C_{15}^2):C_6$ ($C_2^2$), $C_{15}^2:C_{30}$ ($C_4$)

Order 4500: $C_3^2\times C_5^2:D_5.C_2$ ($C_6$), $C_5^3:(C_3^2:C_4)$ ($C_6$), $C_3:S_3\times C_5^2:D_5$ ($C_6$), $C_3\times C_5^3:C_{12}$ ($S_3$)

Order 3375: $C_{15}^2:C_{15}$ ($C_2\times C_4$)

Order 2250: $C_5^3\times C_3:S_3$ ($C_{12}$), $C_3^2\times C_5^2:D_5$ ($C_2\times C_6$), $C_5^3:(C_3:S_3)$ ($C_{12}$), $C_3\times C_5^3:C_6$ ($D_6$)

Order 1500: $C_5^3:C_{12}$ ($C_3\times S_3$)

Order 1350: $C_{15}^2:C_6$ ($F_5$)

Order 1125: $C_5\times C_{15}^2$ ($C_2\times C_{12}$), $C_5^3:C_3^2$ ($C_4\times S_3$)

Order 750: $C_5^3:C_6$ ($C_6\times S_3$)

Order 675: $C_{15}^2:C_3$ ($C_2\times F_5$)

Order 500: $C_5^3:C_4$ ($C_3^2:C_6$)

Order 450: $C_{15}^2:C_2$ ($C_3\times F_5$)

Order 375: $C_5^2\times C_{15}$ ($S_3\times C_{12}$)

Order 250: $C_5^3:C_2$ ($C_3^2:D_6$)

Order 225: $C_{15}^2$ ($C_6\times F_5$), $C_5^2:C_3^2$ ($S_3\times F_5$)

Order 125: $C_5^3$ ($C_4\times C_3^2:C_6$)

Order 90: $C_{15}:S_3$ ($C_5^2:C_{12}$)

Order 75: $C_5\times C_{15}$ ($D_{15}:C_{12}$)

Order 45: $C_3\times C_{15}$ ($C_2\times C_5^2:C_{12}$)

Order 25: $C_5^2$ ($C_3^2:C_6\times F_5$)

Order 18: $C_3:S_3$ ($C_5^3:C_{12}$)

Order 15: $C_{15}$ ($S_3\times C_5^2:C_{12}$)

Order 9: $C_3^2$ ($C_2\times C_5^3:C_{12}$)

Order 5: $C_5$ ($C_{15}^2:(C_2\times C_{12})$)

Order 3: $C_3$ ($S_3\times C_5^3:C_{12}$)

Order 1: $C_1$ ($C_{15}^2:(C_6\times F_5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 27000: $C_{15}^2:(C_6\times F_5)$ ($C_1$)

Order 13500: $(C_5\times C_{15}^2):C_{12}$ ($C_2$), $(C_5\times C_{15}^2):C_{12}$ ($C_2$), $C_{15}^2:(C_3\times D_{10})$ ($C_2$)

Order 9000: $C_5^2:D_5.C_2\times C_3:S_3$ ($C_3$)

Order 6750: $(C_5\times C_{15}^2):C_6$ ($C_4$), $(C_5\times C_{15}^2):C_6$ ($C_2^2$), $C_{15}^2:C_{30}$ ($C_4$)

Order 4500: $C_3^2\times C_5^2:D_5.C_2$ ($C_6$), $C_5^3:(C_3^2:C_4)$ ($C_6$), $C_3:S_3\times C_5^2:D_5$ ($C_6$), $C_3\times C_5^3:C_{12}$ ($S_3$)

Order 3375: $C_{15}^2:C_{15}$ ($C_2\times C_4$)

Order 2250: $C_5^3\times C_3:S_3$ ($C_{12}$), $C_3^2\times C_5^2:D_5$ ($C_2\times C_6$), $C_5^3:(C_3:S_3)$ ($C_{12}$), $C_3\times C_5^3:C_6$ ($D_6$)

Order 1500: $C_5^3:C_{12}$ ($C_3\times S_3$)

Order 1350: $C_{15}^2:C_6$ ($F_5$)

Order 1125: $C_5\times C_{15}^2$ ($C_2\times C_{12}$), $C_5^3:C_3^2$ ($C_4\times S_3$)

Order 750: $C_5^3:C_6$ ($C_6\times S_3$)

Order 675: $C_{15}^2:C_3$ ($C_2\times F_5$)

Order 500: $C_5^3:C_4$ ($C_3^2:C_6$)

Order 450: $C_{15}^2:C_2$ ($C_3\times F_5$)

Order 375: $C_5^2\times C_{15}$ ($S_3\times C_{12}$)

Order 250: $C_5^3:C_2$ ($C_3^2:D_6$)

Order 225: $C_{15}^2$ ($C_6\times F_5$), $C_5^2:C_3^2$ ($S_3\times F_5$)

Order 125: $C_5^3$ ($C_4\times C_3^2:C_6$)

Order 90: $C_{15}:S_3$ ($C_5^2:C_{12}$)

Order 75: $C_5\times C_{15}$ ($D_{15}:C_{12}$)

Order 45: $C_3\times C_{15}$ ($C_2\times C_5^2:C_{12}$)

Order 25: $C_5^2$ ($C_3^2:C_6\times F_5$)

Order 18: $C_3:S_3$ ($C_5^3:C_{12}$)

Order 15: $C_{15}$ ($S_3\times C_5^2:C_{12}$)

Order 9: $C_3^2$ ($C_2\times C_5^3:C_{12}$)

Order 5: $C_5$ ($C_{15}^2:(C_2\times C_{12})$)

Order 3: $C_3$ ($S_3\times C_5^3:C_{12}$)

Order 1: $C_1$ ($C_{15}^2:(C_6\times F_5)$)

Series

Derived series

$C_{15}^2:(C_6\times F_5)$

$\rhd$

$C_{15}^2:(C_6\times F_5)$

$\rhd$

$C_5\times C_{15}^2$

$\rhd$

$C_5\times C_{15}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{15}^2:(C_6\times F_5)$

$\rhd$

$C_{15}^2:(C_6\times F_5)$

$\rhd$

$(C_5\times C_{15}^2):C_{12}$

$\rhd$

$(C_5\times C_{15}^2):C_{12}$

$\rhd$

$(C_5\times C_{15}^2):C_6$

$\rhd$

$(C_5\times C_{15}^2):C_6$

$\rhd$

$C_3^2\times C_5^2:D_5$

$\rhd$

$C_3^2\times C_5^2:D_5$

$\rhd$

$C_5\times C_{15}^2$

$\rhd$

$C_5\times C_{15}^2$

$\rhd$

$C_5^2\times C_{15}$

$\rhd$

$C_5^2\times C_{15}$

$\rhd$

$C_5^3$

$\rhd$

$C_5^3$

$\rhd$

$C_5^2$

$\rhd$

$C_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{15}^2:(C_6\times F_5)$

$\rhd$

$C_{15}^2:(C_6\times F_5)$

$\rhd$

$C_5\times C_{15}^2$

$\rhd$

$C_5\times C_{15}^2$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

See the $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $88 \times 88$ rational character table.