Abstract group 1920.240595: $C_2^2\times D_4\times A_5$

• Label: 1920.240595
• Order: \( 2^{7} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $80$
• Rank: $4$

Group information

Description:	$C_2^2\times D_4\times A_5$
Order:	\(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^4.C_2^4.D_6.S_5$, of order \(368640\)\(\medspace = 2^{13} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 5, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	20
Elements	1	383	20	128	24	460	552	160	192	1920
Conjugacy classes	1	31	1	8	2	15	30	4	8	100
Divisions	1	31	1	8	1	15	15	4	4	80
Autjugacy classes	1	7	1	2	1	3	3	1	1	20

Dimension	1	2	3	4	5	6	8	10	12
Irr. complex chars.	16	4	32	16	16	8	4	4	0	100
Irr. rational chars.	16	4	0	16	16	16	4	4	4	80

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$80$
Rank:	$4$
Inequivalent generating quadruples:	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 $13$ $\langle(1,3)(2,5)(7,8)(10,11)(12,13), (2,5,3)(6,7)(8,9)(10,12)(11,13), (1,2,3)(7,8), (1,2,4,5,3)(6,7)(8,9)\rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $D_4$ $\, \times\, $ $A_5$
Semidirect product:	$(C_2^4\times A_5)$ $\,\rtimes\,$ $C_2$	$(C_4\times A_5)$ $\,\rtimes\,$ $C_2^3$	$C_2^4$ $\,\rtimes\,$ $(C_2\times A_5)$	$C_4$ $\,\rtimes\,$ $(C_2^3\times A_5)$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times A_5)$ . $C_2^4$	$C_2$ . $(C_2^4\times A_5)$	$(C_2^3\times A_5)$ . $C_2^2$	$(C_2^2\times A_5)$ . $C_2^3$	all 6

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 52408 subgroups in 3514 conjugacy classes, 156 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^2\times A_5$
Commutator:	$G' \simeq$ $C_2\times A_5$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^4\times A_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times D_4$	$G/\operatorname{Fit} \simeq$ $A_5$
Radical:	$R \simeq$ $C_2^2\times D_4$	$G/R \simeq$ $A_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times A_5$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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 1920: $C_2^2\times D_4\times A_5$

Order 960: $C_2\times D_4\times A_5$ x 12, $C_2^4\times A_5$ x 2, $C_4\times C_2^2\times A_5$

Order 480: $C_2^3\times A_5$ x 21, $D_4\times A_5$ x 16, $C_2\times C_4\times A_5$ x 6

Order 384: $C_2^6:C_6$

Order 320: $C_{20}:C_2^4$

Order 240: $C_2^2\times A_5$ x 39, $C_4\times A_5$ x 4

Order 192: $C_2^5:C_6$ x 12, $A_4\times C_2^4$ x 2, $C_{12}:C_2^4$, $C_2^4:C_{12}$

Order 160: $D_4\times D_{10}$ x 24, $C_2^3\times D_{10}$ x 2, $C_2^3:D_{10}$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$

Order 128: $D_4\times C_2^4$

Order 120: $C_2\times A_5$ x 15

Order 96: $D_4\times D_6$ x 24, $C_2^3\times A_4$ x 21, $D_4\times A_4$ x 16, $C_2^3:C_{12}$ x 6, $C_2^3\times D_6$ x 2, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 80: $D_4\times D_5$ x 64, $C_2^2\times D_{10}$ x 43, $C_{10}:D_4$ x 24, $D_4\times C_{10}$ x 12, $C_2\times D_{20}$ x 12, $C_4\times D_{10}$ x 12, $C_2^3\times C_{10}$ x 2, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $D_4\times C_2^3$ x 28, $C_2^6$ x 2, $C_2^4\times C_4$

Order 60: $A_5$

Order 48: $S_3\times D_4$ x 64, $C_2^2\times D_6$ x 43, $C_2^2\times A_4$ x 39, $C_6:D_4$ x 24, $C_6\times D_4$ x 12, $C_2\times D_{12}$ x 12, $C_4\times D_6$ x 12, $C_4\times A_4$ x 4, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times D_{10}$ x 162, $C_5:D_4$ x 32, $C_2^2\times C_{10}$ x 21, $D_{20}$ x 16, $C_4\times D_5$ x 16, $C_5\times D_4$ x 16, $C_2\times C_{20}$ x 6, $C_{10}:C_4$ x 6

Order 32: $C_2^2\times D_4$ x 198, $C_2^5$ x 45, $C_2^3\times C_4$ x 14

Order 24: $C_2\times D_6$ x 162, $C_3:D_4$ x 32, $C_2^2\times C_6$ x 21, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_3\times D_4$ x 16, $C_2\times A_4$ x 15, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 20: $D_{10}$ x 124, $C_2\times C_{10}$ x 39, $C_{20}$ x 4, $C_5:C_4$ x 4

Order 16: $C_2\times D_4$ x 328, $C_2^4$ x 291, $C_2^2\times C_4$ x 50

Order 12: $D_6$ x 124, $C_2\times C_6$ x 39, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 10: $D_5$ x 16, $C_{10}$ x 15

Order 8: $C_2^3$ x 529, $D_4$ x 96, $C_2\times C_4$ x 44

Order 6: $S_3$ x 16, $C_6$ x 15

Order 5: $C_5$

Order 4: $C_2^2$ x 249, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1920: $C_2^2\times D_4\times A_5$

Order 960: $C_2^4\times A_5$, $C_2\times D_4\times A_5$, $C_4\times C_2^2\times A_5$

Order 480: $C_2^3\times A_5$ x 3, $D_4\times A_5$, $C_2\times C_4\times A_5$

Order 384: $C_2^6:C_6$

Order 320: $C_{20}:C_2^4$

Order 240: $C_2^2\times A_5$ x 4, $C_4\times A_5$

Order 192: $A_4\times C_2^4$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^4:C_{12}$

Order 160: $D_4\times D_{10}$ x 2, $C_2^3\times D_{10}$, $C_{20}:C_2^3$, $C_2^3:D_{10}$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$

Order 128: $D_4\times C_2^4$

Order 120: $C_2\times A_5$ x 3

Order 96: $C_2^3\times A_4$ x 3, $D_4\times D_6$ x 2, $C_2^3\times D_6$, $C_{12}:C_2^3$, $C_2^3:D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $D_4\times A_4$, $C_2^3:C_{12}$

Order 80: $C_2^2\times D_{10}$ x 6, $D_4\times D_5$ x 2, $C_4\times D_{10}$ x 2, $C_2^3\times C_{10}$, $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_{10}:D_4$, $C_2^2.D_{10}$, $C_2\times D_{20}$

Order 64: $D_4\times C_2^3$ x 5, $C_2^6$, $C_2^4\times C_4$

Order 60: $A_5$

Order 48: $C_2^2\times D_6$ x 6, $C_2^2\times A_4$ x 4, $S_3\times D_4$ x 2, $C_4\times D_6$ x 2, $C_2^3\times C_6$, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_6.C_2^3$, $C_2\times D_{12}$, $C_4\times A_4$

Order 40: $C_2\times D_{10}$ x 11, $C_2^2\times C_{10}$ x 3, $C_4\times D_5$ x 2, $C_2\times C_{20}$, $C_5:D_4$, $C_{10}:C_4$, $D_{20}$, $C_5\times D_4$

Order 32: $C_2^2\times D_4$ x 13, $C_2^5$ x 7, $C_2^3\times C_4$ x 4

Order 24: $C_2\times D_6$ x 11, $C_2^2\times C_6$ x 3, $C_2\times A_4$ x 3, $C_4\times S_3$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_3\times D_4$

Order 20: $D_{10}$ x 10, $C_2\times C_{10}$ x 4, $C_{20}$, $C_5:C_4$

Order 16: $C_2^4$ x 22, $C_2\times D_4$ x 10, $C_2^2\times C_4$ x 9

Order 12: $D_6$ x 10, $C_2\times C_6$ x 4, $A_4$, $C_{12}$, $C_3:C_4$

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

Order 8: $C_2^3$ x 31, $C_2\times C_4$ x 6, $D_4$ x 4

Order 6: $S_3$ x 4, $C_6$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 22, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1920: $C_2^2\times D_4\times A_5$ ($C_1$)

Order 960: $C_2\times D_4\times A_5$ ($C_2$) x 12, $C_2^4\times A_5$ ($C_2$) x 2, $C_4\times C_2^2\times A_5$ ($C_2$)

Order 480: $D_4\times A_5$ ($C_2^2$) x 16, $C_2^3\times A_5$ ($C_2^2$) x 13, $C_2\times C_4\times A_5$ ($C_2^2$) x 6

Order 240: $C_2^2\times A_5$ ($C_2^3$) x 11, $C_4\times A_5$ ($C_2^3$) x 4, $C_2^2\times A_5$ ($D_4$) x 4

Order 120: $C_2\times A_5$ ($C_2\times D_4$) x 6, $C_2\times A_5$ ($C_2^4$)

Order 60: $A_5$ ($C_2^2\times D_4$)

Order 32: $C_2^2\times D_4$ ($A_5$)

Order 16: $C_2\times D_4$ ($C_2\times A_5$) x 12, $C_2^4$ ($C_2\times A_5$) x 2, $C_2^2\times C_4$ ($C_2\times A_5$)

Order 8: $D_4$ ($C_2^2\times A_5$) x 16, $C_2^3$ ($C_2^2\times A_5$) x 13, $C_2\times C_4$ ($C_2^2\times A_5$) x 6

Order 4: $C_2^2$ ($C_2^3\times A_5$) x 11, $C_2^2$ ($D_4\times A_5$) x 4, $C_4$ ($C_2^3\times A_5$) x 4

Order 2: $C_2$ ($C_2\times D_4\times A_5$) x 6, $C_2$ ($C_2^4\times A_5$)

Order 1: $C_1$ ($C_2^2\times D_4\times A_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1920: $C_2^2\times D_4\times A_5$ ($C_1$)

Order 960: $C_2^4\times A_5$ ($C_2$), $C_2\times D_4\times A_5$ ($C_2$), $C_4\times C_2^2\times A_5$ ($C_2$)

Order 480: $C_2^3\times A_5$ ($C_2^2$) x 2, $D_4\times A_5$ ($C_2^2$), $C_2\times C_4\times A_5$ ($C_2^2$)

Order 240: $C_2^2\times A_5$ ($C_2^3$) x 2, $C_4\times A_5$ ($C_2^3$), $C_2^2\times A_5$ ($D_4$)

Order 120: $C_2\times A_5$ ($C_2^4$), $C_2\times A_5$ ($C_2\times D_4$)

Order 60: $A_5$ ($C_2^2\times D_4$)

Order 32: $C_2^2\times D_4$ ($A_5$)

Order 16: $C_2^4$ ($C_2\times A_5$), $C_2\times D_4$ ($C_2\times A_5$), $C_2^2\times C_4$ ($C_2\times A_5$)

Order 8: $C_2^3$ ($C_2^2\times A_5$) x 2, $D_4$ ($C_2^2\times A_5$), $C_2\times C_4$ ($C_2^2\times A_5$)

Order 4: $C_2^2$ ($C_2^3\times A_5$) x 2, $C_2^2$ ($D_4\times A_5$), $C_4$ ($C_2^3\times A_5$)

Order 2: $C_2$ ($C_2^4\times A_5$), $C_2$ ($C_2\times D_4\times A_5$)

Order 1: $C_1$ ($C_2^2\times D_4\times A_5$)

Series

Derived series

$C_2^2\times D_4\times A_5$

$\rhd$

$C_2\times A_5$

$\rhd$

$A_5$

Chief series

$C_2^2\times D_4\times A_5$

$\rhd$

$C_2^2\times D_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^2\times D_4\times A_5$

$\rhd$

$C_2\times A_5$

$\rhd$

$A_5$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times D_4$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 5 larger groups in the database.

Character theory

Complex character table

See the $100 \times 100$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $80 \times 80$ rational character table.