Abstract group 2880.gn: $C_2^4:\GL(2,4)$

• Label: 2880.gn
• Order: \( 2^{6} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $60$
• Rank: $2$

Group information

Description:	$C_2^4:\GL(2,4)$
Order:	\(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \)
Exponent:	\(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism group:	$S_3\times S_4\times S_5$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$, $A_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	5	6	10	15	30
Elements	1	255	188	24	1284	360	192	576	2880
Conjugacy classes	1	15	5	2	27	14	4	12	80
Divisions	1	15	3	1	17	7	1	3	48
Autjugacy classes	1	7	3	1	7	3	1	1	24

Dimension	1	2	3	4	5	6	8	9	10	12	15	18
Irr. complex chars.	12	0	28	12	12	0	0	8	0	4	4	0	80
Irr. rational chars.	4	4	4	4	4	4	4	0	4	8	4	4	48

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$60$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $13$ $\langle(8,9)(12,13), (6,12,8,7,13,9), (6,7)(12,13), (1,4,5)(6,12,8,7,13,9)(10,11), (1,2)(3,4)(10,11)(12,13), (8,9)(10,11)(12,13), (8,9)\rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $A_4$ $\, \times\, $ $A_5$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $\GL(2,4)$	$(C_2^4\times A_5)$ $\,\rtimes\,$ $C_3$	$(C_2^3\times A_5)$ $\,\rtimes\,$ $C_6$	$C_2^3$ $\,\rtimes\,$ $(C_6\times A_5)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_2^7.S_4)$	$\Aut((C_2^2\times S_4^2).D_4)$	$\Aut(C_3^6.S_3^2)$	$\Aut(C_{219}\times \SL(2,5))$

Elements of the group are displayed as permutations of degree 13.

Homology

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

Subgroups

There are 24598 subgroups in 921 conjugacy classes, 30 normal (12 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $A_4\times A_5$
Commutator:	$G' \simeq$ $C_2^2\times A_5$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^4:\GL(2,4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4$	$G/\operatorname{Fit} \simeq$ $\GL(2,4)$
Radical:	$R \simeq$ $C_2^2\times A_4$	$G/R \simeq$ $A_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times A_5$	$G/\operatorname{soc} \simeq$ $C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 2880: $C_2^4:\GL(2,4)$

Order 1440: $C_2^3:\GL(2,4)$ x 3

Order 960: $C_2^4\times A_5$

Order 720: $C_2^2\times \GL(2,4)$, $A_4\times A_5$

Order 576: $C_2^2\times A_4^2$

Order 480: $C_2^3\times A_5$ x 7, $C_2\times A_4\times D_{10}$

Order 360: $C_6\times A_5$ x 3

Order 288: $C_2\times A_4^2$ x 3, $C_2\times A_4\times D_6$

Order 240: $C_2^2\times A_5$ x 13, $A_4\times D_{10}$ x 6, $C_2^3:C_{30}$

Order 192: $C_2^2\wr C_3$ x 2, $A_4\times C_2^4$ x 2

Order 180: $\GL(2,4)$

Order 160: $C_2^3\times D_{10}$

Order 144: $A_4\times D_6$ x 6, $C_2^2:C_6^2$ x 2, $A_4^2$

Order 120: $C_2\times A_5$ x 7, $D_5\times A_4$ x 4, $C_{10}\times A_4$ x 3, $C_6\times D_{10}$

Order 96: $C_2^3\times A_4$ x 14, $C_2^3:A_4$ x 6, $C_2^3\times D_6$

Order 80: $C_2^2\times D_{10}$ x 14, $C_2^3\times C_{10}$

Order 72: $C_6\times A_4$ x 6, $S_3\times A_4$ x 4, $C_6\times D_6$

Order 64: $C_2^6$

Order 60: $C_3\times D_{10}$ x 6, $C_5\times A_4$, $A_5$, $C_2\times C_{30}$

Order 48: $C_2^2\times A_4$ x 31, $C_2^2\times D_6$ x 14, $C_2^3\times C_6$ x 2, $C_2^2:A_4$ x 2

Order 40: $C_2\times D_{10}$ x 50, $C_2^2\times C_{10}$ x 7

Order 36: $C_6\times S_3$ x 6, $C_3\times A_4$ x 2, $C_6^2$

Order 32: $C_2^5$ x 15

Order 30: $C_3\times D_5$ x 4, $C_{30}$ x 3

Order 24: $C_2\times D_6$ x 50, $C_2\times A_4$ x 29, $C_2^2\times C_6$ x 14

Order 20: $D_{10}$ x 44, $C_2\times C_{10}$ x 13

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2^4$ x 91

Order 15: $C_{15}$

Order 12: $D_6$ x 44, $C_2\times C_6$ x 27, $A_4$ x 7

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 9: $C_3^2$

Order 8: $C_2^3$ x 175

Order 6: $C_6$ x 17, $S_3$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 91

Order 3: $C_3$ x 3

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2880: $C_2^4:\GL(2,4)$

Order 1440: $C_2^3:\GL(2,4)$

Order 960: $C_2^4\times A_5$

Order 720: $C_2^2\times \GL(2,4)$, $A_4\times A_5$

Order 576: $C_2^2\times A_4^2$

Order 480: $C_2^3\times A_5$ x 3, $C_2\times A_4\times D_{10}$

Order 360: $C_6\times A_5$

Order 288: $C_2\times A_4\times D_6$, $C_2\times A_4^2$

Order 240: $C_2^2\times A_5$ x 6, $A_4\times D_{10}$ x 2, $C_2^3:C_{30}$

Order 192: $A_4\times C_2^4$ x 2, $C_2^2\wr C_3$

Order 180: $\GL(2,4)$

Order 160: $C_2^3\times D_{10}$

Order 144: $C_2^2:C_6^2$ x 2, $A_4\times D_6$ x 2, $A_4^2$

Order 120: $C_2\times A_5$ x 3, $D_5\times A_4$ x 2, $C_6\times D_{10}$, $C_{10}\times A_4$

Order 96: $C_2^3\times A_4$ x 6, $C_2^3\times D_6$, $C_2^3:A_4$

Order 80: $C_2^2\times D_{10}$ x 6, $C_2^3\times C_{10}$

Order 72: $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_6\times D_6$

Order 64: $C_2^6$

Order 60: $C_3\times D_{10}$ x 2, $C_5\times A_4$, $A_5$, $C_2\times C_{30}$

Order 48: $C_2^2\times A_4$ x 15, $C_2^2\times D_6$ x 6, $C_2^3\times C_6$ x 2, $C_2^2:A_4$

Order 40: $C_2\times D_{10}$ x 16, $C_2^2\times C_{10}$ x 3

Order 36: $C_6\times S_3$ x 2, $C_3\times A_4$ x 2, $C_6^2$

Order 32: $C_2^5$ x 7

Order 30: $C_3\times D_5$ x 2, $C_{30}$

Order 24: $C_2\times D_6$ x 16, $C_2\times A_4$ x 9, $C_2^2\times C_6$ x 6

Order 20: $D_{10}$ x 13, $C_2\times C_{10}$ x 6

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

Order 16: $C_2^4$ x 29

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 13, $D_6$ x 13, $A_4$ x 5

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $C_2^3$ x 42

Order 6: $C_6$ x 7, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 29

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2880: $C_2^4:\GL(2,4)$ ($C_1$)

Order 1440: $C_2^3:\GL(2,4)$ ($C_2$) x 3

Order 960: $C_2^4\times A_5$ ($C_3$)

Order 720: $A_4\times A_5$ ($C_2^2$)

Order 480: $C_2^3\times A_5$ ($C_6$) x 3

Order 240: $C_2^2\times A_5$ ($C_2\times C_6$), $C_2^2\times A_5$ ($A_4$)

Order 120: $C_2\times A_5$ ($C_2\times A_4$) x 3

Order 60: $A_5$ ($C_2^2\times A_4$)

Order 48: $C_2^2\times A_4$ ($A_5$)

Order 24: $C_2\times A_4$ ($C_2\times A_5$) x 3

Order 16: $C_2^4$ ($\GL(2,4)$)

Order 12: $A_4$ ($C_2^2\times A_5$)

Order 8: $C_2^3$ ($C_6\times A_5$) x 3

Order 4: $C_2^2$ ($C_2^2\times \GL(2,4)$), $C_2^2$ ($A_4\times A_5$)

Order 2: $C_2$ ($C_2^3:\GL(2,4)$) x 3

Order 1: $C_1$ ($C_2^4:\GL(2,4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2880: $C_2^4:\GL(2,4)$ ($C_1$)

Order 1440: $C_2^3:\GL(2,4)$ ($C_2$)

Order 960: $C_2^4\times A_5$ ($C_3$)

Order 720: $A_4\times A_5$ ($C_2^2$)

Order 480: $C_2^3\times A_5$ ($C_6$)

Order 240: $C_2^2\times A_5$ ($C_2\times C_6$), $C_2^2\times A_5$ ($A_4$)

Order 120: $C_2\times A_5$ ($C_2\times A_4$)

Order 60: $A_5$ ($C_2^2\times A_4$)

Order 48: $C_2^2\times A_4$ ($A_5$)

Order 24: $C_2\times A_4$ ($C_2\times A_5$)

Order 16: $C_2^4$ ($\GL(2,4)$)

Order 12: $A_4$ ($C_2^2\times A_5$)

Order 8: $C_2^3$ ($C_6\times A_5$)

Order 4: $C_2^2$ ($C_2^2\times \GL(2,4)$), $C_2^2$ ($A_4\times A_5$)

Order 2: $C_2$ ($C_2^3:\GL(2,4)$)

Order 1: $C_1$ ($C_2^4:\GL(2,4)$)

Series

Derived series

$C_2^4:\GL(2,4)$

$\rhd$

$C_2^2\times A_5$

$\rhd$

$A_5$

Chief series

$C_2^4:\GL(2,4)$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^4:\GL(2,4)$

$\rhd$

$C_2^2\times A_5$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 20 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.