Abstract group 6480.cx: $C_3^2:C_{12}\times A_5$

• Label: 6480.cx
• Order: \( 2^{4} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $14$
• Trans deg.: $60$
• Rank: $2$

Group information

Description:	$C_3^2:C_{12}\times A_5$
Order:	\(6480\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2\times F_9.(C_2\times S_5)$, of order \(34560\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$ x 3, $A_5$
Derived length:	$2$

This group is nonabelian, an A-group, and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	159	566	288	24	1218	216	1656	624	432	432	864	6480
Conjugacy classes	1	3	17	4	2	15	2	14	16	4	4	8	90
Divisions	1	3	11	2	1	9	1	4	5	1	1	1	40
Autjugacy classes	1	3	7	2	1	7	1	4	3	1	1	1	32

Dimension	1	2	3	4	5	6	8	10	12	16	20	24	32	40	48
Irr. complex chars.	12	0	24	18	12	0	0	0	12	6	6	0	0	0	0	90
Irr. rational chars.	2	3	0	5	2	2	5	3	3	3	3	3	2	2	2	40

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$60$
Rank:	$2$
Inequivalent generating pairs:	$456$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	32
Arbitrary	7	9	10

Constructions

Permutation group:	Degree $14$ $\langle(10,12,11), (6,13,14)(10,11,12), (1,4,5)(6,14,13)(7,8,9)(10,12,11), (7,9,8) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_3^2:C_4)$ $\, \times\, $ $A_5$
Semidirect product:	$(C_3^3\times A_5)$ $\,\rtimes\,$ $C_4$	$C_3^3$ $\,\rtimes\,$ $(C_4\times A_5)$	$C_3^2$ $\,\rtimes\,$ $(C_{12}\times A_5)$	$(C_3\times \GL(2,4))$ $\,\rtimes\,$ $C_{12}$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(\GL(2,4):S_3)$ . $C_6$	$(C_3:S_3)$ . $(C_6\times A_5)$	$(C_3^2:C_6)$ . $(C_2\times A_5)$	$(C_3:S_3\times \GL(2,4))$ . $C_2$	more information

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 13592 subgroups in 360 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^2:C_4\times A_5$
Commutator:	$G' \simeq$ $C_3\times \GL(2,4)$	$G/G' \simeq$ $C_{12}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^2:C_{12}\times A_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3$	$G/\operatorname{Fit} \simeq$ $C_4\times A_5$
Radical:	$R \simeq$ $C_3^2:C_{12}$	$G/R \simeq$ $A_5$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times A_5$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 6480: $C_3^2:C_{12}\times A_5$

Order 3240: $C_3:S_3\times \GL(2,4)$

Order 2160: $C_3^2:C_4\times A_5$

Order 1620: $C_3^3\times A_5$

Order 1296: $C_3\times C_6^2:C_{12}$

Order 1080: $S_3\times \GL(2,4)$ x 2, $C_3^2:C_{12}\times D_5$, $\GL(2,4):S_3$

Order 720: $C_{12}\times A_5$

Order 648: $C_3\times C_6^2:C_6$, $S_3\times C_3^2:C_{12}$

Order 540: $C_3\times \GL(2,4)$ x 5, $(C_3\times C_{15}):C_{12}$, $C_3^2:C_{60}$, $C_3^3:D_{10}$

Order 432: $C_6^2:C_{12}$ x 3, $C_6^2:C_{12}$

Order 360: $S_3\times A_5$ x 2, $C_3^2:C_4\times D_5$, $C_6\times A_5$

Order 324: $A_4\times C_3^3$, $C_3^2:S_3^2$, $C_3\wr C_4$, $C_3^3:C_{12}$

Order 270: $C_3^2:D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$

Order 240: $C_4\times A_5$

Order 216: $C_6^2:C_6$ x 3, $C_2\times C_3^2:C_{12}$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_3^2:C_4\times S_3$

Order 180: $\GL(2,4)$ x 5, $C_{15}:D_6$ x 2, $C_{15}:D_6$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^3:C_6$, $C_3^2\wr C_2$, $S_3\times C_3^3$

Order 144: $C_6^2:C_4$, $C_{12}\times A_4$

Order 135: $C_3^2\times C_{15}$

Order 120: $C_2\times A_5$, $D_5\times C_{12}$

Order 108: $C_3^2\times A_4$ x 13, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 2, $C_3\times S_3^2$ x 2, $C_3\times C_6^2$, $C_3:S_3^2$, $C_3^3:C_4$

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

Order 81: $C_3^4$

Order 72: $S_3\times A_4$ x 6, $C_6\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_6:D_6$, $C_6\times A_4$, $S_3\times C_{12}$

Order 60: $S_3\times D_5$ x 2, $A_5$, $C_{60}$, $C_5:C_{12}$, $C_3\times D_{10}$

Order 54: $C_3^2:C_6$ x 8, $S_3\times C_3^2$ x 7, $C_3^2\times C_6$, $C_3^2:S_3$

Order 48: $C_4\times A_4$ x 3, $C_2^2\times C_{12}$

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

Order 40: $C_4\times D_5$

Order 36: $C_3\times A_4$ x 33, $C_6^2$ x 5, $C_6\times S_3$ x 5, $C_3^2:C_4$ x 2, $C_6:S_3$ x 2, $S_3^2$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$

Order 30: $C_3\times D_5$ x 6, $D_{15}$ x 2, $C_5\times S_3$ x 2, $C_{30}$

Order 27: $C_3^3$ x 11

Order 24: $C_2\times A_4$ x 3, $C_2\times C_{12}$ x 2, $C_2\times D_6$ x 2, $C_4\times S_3$, $C_2^2\times C_6$

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 16, $C_3\times C_6$ x 6, $C_3:S_3$ x 6

Order 16: $C_2^2\times C_4$

Order 15: $C_{15}$ x 5

Order 12: $A_4$ x 9, $C_2\times C_6$ x 7, $D_6$ x 5, $C_{12}$ x 4, $C_3:C_4$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 26

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

Order 6: $C_6$ x 9, $S_3$ x 8

Order 5: $C_5$

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

Order 3: $C_3$ x 11

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6480: $C_3^2:C_{12}\times A_5$

Order 3240: $C_3:S_3\times \GL(2,4)$

Order 2160: $C_3^2:C_4\times A_5$

Order 1620: $C_3^3\times A_5$

Order 1296: $C_3\times C_6^2:C_{12}$

Order 1080: $C_3^2:C_{12}\times D_5$, $S_3\times \GL(2,4)$, $\GL(2,4):S_3$

Order 720: $C_{12}\times A_5$

Order 648: $C_3\times C_6^2:C_6$, $S_3\times C_3^2:C_{12}$

Order 540: $C_3\times \GL(2,4)$ x 3, $(C_3\times C_{15}):C_{12}$, $C_3^2:C_{60}$, $C_3^3:D_{10}$

Order 432: $C_6^2:C_{12}$ x 2, $C_6^2:C_{12}$

Order 360: $C_3^2:C_4\times D_5$, $C_6\times A_5$, $S_3\times A_5$

Order 324: $A_4\times C_3^3$, $C_3^2:S_3^2$, $C_3\wr C_4$, $C_3^3:C_{12}$

Order 270: $C_3^2:D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$

Order 240: $C_4\times A_5$

Order 216: $C_2\times C_3^2:C_{12}$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:C_4\times S_3$

Order 180: $\GL(2,4)$ x 3, $C_{15}:D_6$, $C_{15}:D_6$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^3:C_6$, $C_3^2\wr C_2$, $S_3\times C_3^3$

Order 144: $C_6^2:C_4$, $C_{12}\times A_4$

Order 135: $C_3^2\times C_{15}$

Order 120: $C_2\times A_5$, $D_5\times C_{12}$

Order 108: $C_3^2\times A_4$ x 6, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 2, $C_3\times C_6^2$, $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^3:C_4$

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

Order 81: $C_3^4$

Order 72: $C_2\times C_3^2:C_4$ x 2, $S_3\times A_4$ x 2, $C_6:D_6$, $C_6\times D_6$, $C_6\times A_4$, $S_3\times C_{12}$

Order 60: $S_3\times D_5$, $A_5$, $C_{60}$, $C_5:C_{12}$, $C_3\times D_{10}$

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 4, $C_3^2\times C_6$, $C_3^2:S_3$

Order 48: $C_4\times A_4$ x 2, $C_2^2\times C_{12}$

Order 45: $C_3\times C_{15}$ x 3

Order 40: $C_4\times D_5$

Order 36: $C_3\times A_4$ x 10, $C_6^2$ x 3, $C_6\times S_3$ x 3, $C_3^2:C_4$ x 2, $C_6:S_3$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$, $S_3^2$

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

Order 27: $C_3^3$ x 7

Order 24: $C_2\times C_{12}$ x 2, $C_2\times A_4$ x 2, $C_4\times S_3$, $C_2^2\times C_6$, $C_2\times D_6$

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 9, $C_3\times C_6$ x 4, $C_3:S_3$ x 4

Order 16: $C_2^2\times C_4$

Order 15: $C_{15}$ x 3

Order 12: $C_2\times C_6$ x 5, $A_4$ x 4, $C_{12}$ x 4, $D_6$ x 3, $C_3:C_4$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 13

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

Order 6: $C_6$ x 7, $S_3$ x 5

Order 5: $C_5$

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6480: $C_3^2:C_{12}\times A_5$ ($C_1$)

Order 3240: $C_3:S_3\times \GL(2,4)$ ($C_2$)

Order 2160: $C_3^2:C_4\times A_5$ ($C_3$)

Order 1620: $C_3^3\times A_5$ ($C_4$)

Order 1080: $\GL(2,4):S_3$ ($C_6$)

Order 540: $C_3\times \GL(2,4)$ ($C_{12}$)

Order 180: $\GL(2,4)$ ($C_3^2:C_4$)

Order 108: $C_3^2:C_{12}$ ($A_5$)

Order 60: $A_5$ ($C_3^2:C_{12}$)

Order 54: $C_3^2:C_6$ ($C_2\times A_5$)

Order 36: $C_3^2:C_4$ ($\GL(2,4)$)

Order 27: $C_3^3$ ($C_4\times A_5$)

Order 18: $C_3:S_3$ ($C_6\times A_5$)

Order 9: $C_3^2$ ($C_{12}\times A_5$)

Order 3: $C_3$ ($C_3^2:C_4\times A_5$)

Order 1: $C_1$ ($C_3^2:C_{12}\times A_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6480: $C_3^2:C_{12}\times A_5$ ($C_1$)

Order 3240: $C_3:S_3\times \GL(2,4)$ ($C_2$)

Order 2160: $C_3^2:C_4\times A_5$ ($C_3$)

Order 1620: $C_3^3\times A_5$ ($C_4$)

Order 1080: $\GL(2,4):S_3$ ($C_6$)

Order 540: $C_3\times \GL(2,4)$ ($C_{12}$)

Order 180: $\GL(2,4)$ ($C_3^2:C_4$)

Order 108: $C_3^2:C_{12}$ ($A_5$)

Order 60: $A_5$ ($C_3^2:C_{12}$)

Order 54: $C_3^2:C_6$ ($C_2\times A_5$)

Order 36: $C_3^2:C_4$ ($\GL(2,4)$)

Order 27: $C_3^3$ ($C_4\times A_5$)

Order 18: $C_3:S_3$ ($C_6\times A_5$)

Order 9: $C_3^2$ ($C_{12}\times A_5$)

Order 3: $C_3$ ($C_3^2:C_4\times A_5$)

Order 1: $C_1$ ($C_3^2:C_{12}\times A_5$)

Series

Derived series

$C_3^2:C_{12}\times A_5$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$A_5$

Chief series

$C_3^2:C_{12}\times A_5$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3^2:C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_3^2:C_{12}\times A_5$

$\rhd$

$C_3\times \GL(2,4)$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.