Abstract group 1920.240594: $C_2\times D_4\times S_5$

• Label: 1920.240594
• Order: \( 2^{7} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $11$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$C_2\times D_4\times S_5$
Order:	\(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^5.C_2^3.S_5$, of order \(30720\)\(\medspace = 2^{11} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 5, $A_5$
Derived length:	$2$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	10	12	20
Elements	1	311	20	584	24	460	264	160	96	1920
Conjugacy classes	1	23	1	16	1	15	7	4	2	70
Divisions	1	23	1	16	1	15	7	4	2	70
Autjugacy classes	1	9	1	6	1	5	3	2	1	29

Dimension	1	2	4	5	6	8	10	12
Irr. complex chars.	16	4	16	16	8	4	4	2	70
Irr. rational chars.	16	4	16	16	8	4	4	2	70

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$40$
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 $11$ $\langle(1,4,3,5,2)(7,8), (1,2)(3,4,5)(6,7,9,8), (1,3)(2,5)(6,8)(7,9), (2,3,5,4)(7,8)(10,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Transitive group:	40T1626				more information
Direct product:	$C_2$ $\, \times\, $ $D_4$ $\, \times\, $ $S_5$
Semidirect product:	$(C_2^3\times S_5)$ $\,\rtimes\,$ $C_2$	$(C_2^3:S_5)$ $\,\rtimes\,$ $C_2$	$(C_4\times S_5)$ $\,\rtimes\,$ $C_2^2$	$(C_4:S_5)$ $\,\rtimes\,$ $C_2^2$	all 23
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times S_5)$ . $C_2^3$	$C_2$ . $(C_2^3\times S_5)$	$(C_2\times A_5)$ . $C_2^4$	$(C_2^2\times S_5)$ . $C_2^2$	all 6
Aut. group:	$\Aut(C_8:S_5)$	$\Aut(C_8.S_5)$	$\Aut(A_5:\OD_{16})$	$\Aut(C_2\times A_5:C_8)$	all 37

Elements of the group are displayed as permutations of degree 11.

Homology

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

Subgroups

There are 36502 subgroups in 2189 conjugacy classes, 97 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times S_5$
Commutator:	$G' \simeq$ $C_2\times A_5$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times D_4$	$G/\operatorname{Fit} \simeq$ $S_5$
Radical:	$R \simeq$ $C_2\times D_4$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times A_5$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4^2$
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

No diagram available: subgroups only stored up to automorphism

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: $C_2\times D_4\times S_5$

Order 960: $D_4\times S_5$ x 8, $C_2^3\times S_5$ x 2, $C_2^3:S_5$ x 2, $C_2\times D_4\times A_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 480: $C_2^2\times S_5$ x 19, $C_2^2:S_5$ x 8, $D_4\times A_5$ x 4, $C_4:S_5$ x 4, $C_4\times S_5$ x 4, $C_2^3\times A_5$ x 2, $C_2\times C_4\times A_5$, $C_2^2.S_5$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$

Order 320: $D_{10}.C_2^4$

Order 240: $C_2\times S_5$ x 30, $C_2^2\times A_5$ x 9, $C_4\times A_5$ x 2, $A_5:C_4$ x 2

Order 192: $D_4\times S_4$ x 8, $C_2^3\times S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 160: $D_4\times F_5$ x 8, $C_2^3\times F_5$ x 2, $C_2^3:F_5$ x 2, $D_4\times D_{10}$, $D_{10}.D_4$, $C_{10}:C_4^2$

Order 128: $C_2\times D_4^2$

Order 120: $S_5$ x 8, $C_2\times A_5$ x 7

Order 96: $D_4\times D_6$ x 24, $C_2^2\times S_4$ x 19, $\GL(2,\mathbb{Z}/4)$ x 8, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $C_4\times S_4$ x 4, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 80: $C_2^2\times F_5$ x 20, $D_4\times D_5$ x 8, $D_{10}:C_4$ x 8, $C_{20}:C_4$ x 4, $C_4\times F_5$ x 4, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 64: $D_4^2$ x 16, $D_4\times C_2^3$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$

Order 60: $A_5$

Order 48: $S_3\times D_4$ x 64, $C_2^2\times D_6$ x 43, $C_2\times S_4$ x 30, $C_6:D_4$ x 24, $C_6\times D_4$ x 12, $C_2\times D_{12}$ x 12, $C_4\times D_6$ x 12, $C_2^2\times A_4$ x 9, $C_2^3\times C_6$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times F_5$ x 34, $C_2\times D_{10}$ x 19, $C_5:D_4$ x 8, $D_{20}$ x 4, $C_4\times D_5$ x 4, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 32: $C_2^2\times D_4$ x 72, $C_4:D_4$ x 32, $C_2^2\wr C_2$ x 32, $C_4\times D_4$ x 16, $C_4:D_4$ x 8, $C_2^3:C_4$ x 8, $C_2^5$ x 4, $C_2^3\times C_4$ x 4, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

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, $S_4$ x 8, $C_2\times A_4$ x 7, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 20: $D_{10}$ x 30, $F_5$ x 10, $C_2\times C_{10}$ x 9, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $C_2\times D_4$ x 223, $C_2^4$ x 52, $C_2^2\times C_4$ x 35, $C_2^2:C_4$ x 32, $C_4:C_4$ x 8, $C_4^2$ x 4

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 8, $C_{10}$ x 7

Order 8: $C_2^3$ x 159, $D_4$ x 116, $C_2\times C_4$ x 55

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

Order 5: $C_5$

Order 4: $C_2^2$ x 124, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1920: $C_2\times D_4\times S_5$

Order 960: $C_2^3\times S_5$, $C_2\times D_4\times A_5$, $C_2^3:S_5$, $D_4\times S_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 480: $C_2^2\times S_5$ x 4, $D_4\times A_5$, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $C_2^2:S_5$, $C_4:S_5$, $C_4\times S_5$, $C_2^3\times A_5$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$

Order 320: $D_{10}.C_2^4$

Order 240: $C_2\times S_5$ x 5, $C_2^2\times A_5$ x 3, $C_4\times A_5$, $A_5:C_4$

Order 192: $C_2^3\times S_4$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $D_4\times S_4$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 160: $C_2^3\times F_5$, $D_4\times D_{10}$, $C_2^3:F_5$, $D_4\times F_5$, $D_{10}.D_4$, $C_{10}:C_4^2$

Order 128: $C_2\times D_4^2$

Order 120: $C_2\times A_5$ x 3, $S_5$ x 2

Order 96: $D_4\times D_6$ x 8, $C_2^2\times S_4$ x 4, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^3:D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $D_4\times A_4$, $C_2^3:C_{12}$, $\GL(2,\mathbb{Z}/4)$, $C_2^2.S_4$, $C_4:S_4$, $C_4\times S_4$

Order 80: $C_2^2\times F_5$ x 5, $D_4\times D_5$ x 2, $C_2^2\times D_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $C_2\times D_{20}$, $C_4\times D_{10}$, $D_{10}:C_4$, $C_{20}:C_4$, $C_4\times F_5$

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

Order 60: $A_5$

Order 48: $C_2^2\times D_6$ x 10, $S_3\times D_4$ x 9, $C_4\times D_6$ x 6, $C_2\times S_4$ x 5, $C_6:D_4$ x 5, $C_6\times D_4$ x 4, $C_2\times D_{12}$ x 4, $C_2^2\times A_4$ x 3, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_4\times A_4$, $A_4:C_4$

Order 40: $C_2\times F_5$ x 7, $C_2\times D_{10}$ x 6, $C_4\times D_5$ x 2, $C_2\times C_{20}$, $C_5:D_4$, $C_{10}:C_4$, $D_{20}$, $C_2^2\times C_{10}$, $C_5\times D_4$

Order 32: $C_2^2\times D_4$ x 21, $C_2^3:C_4$ x 5, $C_4:D_4$ x 4, $C_2^3\times C_4$ x 3, $C_2^2\wr C_2$ x 3, $C_4\times D_4$ x 3, $C_2^5$ x 2, $C_2^2.D_4$ x 2, $C_4:D_4$, $C_2\times C_4^2$

Order 24: $C_2\times D_6$ x 25, $C_4\times S_3$ x 6, $C_2^2\times C_6$ x 5, $C_3:D_4$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $D_{12}$ x 3, $C_2\times A_4$ x 3, $C_3\times D_4$ x 3, $S_4$ x 2

Order 20: $D_{10}$ x 9, $C_2\times C_{10}$ x 3, $F_5$ x 3, $C_{20}$, $C_5:C_4$

Order 16: $C_2\times D_4$ x 38, $C_2^4$ x 15, $C_2^2\times C_4$ x 14, $C_2^2:C_4$ x 6, $C_4:C_4$ x 2, $C_4^2$

Order 12: $D_6$ x 22, $C_2\times C_6$ x 8, $C_{12}$ x 2, $C_3:C_4$ x 2, $A_4$

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

Order 8: $C_2^3$ x 36, $C_2\times C_4$ x 18, $D_4$ x 17

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

Order 5: $C_5$

Order 4: $C_2^2$ x 31, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1920: $C_2\times D_4\times S_5$ ($C_1$)

Order 960: $D_4\times S_5$ ($C_2$) x 8, $C_2^3\times S_5$ ($C_2$) x 2, $C_2^3:S_5$ ($C_2$) x 2, $C_2\times D_4\times A_5$ ($C_2$), $C_2\times C_4:S_5$ ($C_2$), $C_2\times C_4\times S_5$ ($C_2$)

Order 480: $C_2^2\times S_5$ ($C_2^2$) x 11, $C_2^2:S_5$ ($C_2^2$) x 8, $D_4\times A_5$ ($C_2^2$) x 4, $C_4:S_5$ ($C_2^2$) x 4, $C_4\times S_5$ ($C_2^2$) x 4, $C_2^3\times A_5$ ($C_2^2$) x 2, $C_2\times C_4\times A_5$ ($C_2^2$), $C_2^2.S_5$ ($C_2^2$)

Order 240: $C_2\times S_5$ ($C_2^3$) x 6, $C_2^2\times A_5$ ($C_2^3$) x 5, $C_2\times S_5$ ($D_4$) x 4, $C_4\times A_5$ ($C_2^3$) x 2, $A_5:C_4$ ($C_2^3$) x 2

Order 120: $S_5$ ($C_2\times D_4$) x 4, $C_2\times A_5$ ($C_2\times D_4$) x 2, $C_2\times A_5$ ($C_2^4$)

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

Order 16: $C_2\times D_4$ ($S_5$)

Order 8: $D_4$ ($C_2\times S_5$) x 4, $C_2^3$ ($C_2\times S_5$) x 2, $C_2\times C_4$ ($C_2\times S_5$)

Order 4: $C_2^2$ ($C_2^2\times S_5$) x 5, $C_4$ ($C_2^2\times S_5$) x 2

Order 2: $C_2$ ($D_4\times S_5$) x 2, $C_2$ ($C_2^3\times S_5$)

Order 1: $C_1$ ($C_2\times D_4\times S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1920: $C_2\times D_4\times S_5$ ($C_1$)

Order 960: $C_2^3\times S_5$ ($C_2$), $C_2\times D_4\times A_5$ ($C_2$), $C_2^3:S_5$ ($C_2$), $D_4\times S_5$ ($C_2$), $C_2\times C_4:S_5$ ($C_2$), $C_2\times C_4\times S_5$ ($C_2$)

Order 480: $C_2^2\times S_5$ ($C_2^2$) x 3, $D_4\times A_5$ ($C_2^2$), $C_2\times C_4\times A_5$ ($C_2^2$), $C_2^2.S_5$ ($C_2^2$), $C_2^2:S_5$ ($C_2^2$), $C_4:S_5$ ($C_2^2$), $C_4\times S_5$ ($C_2^2$), $C_2^3\times A_5$ ($C_2^2$)

Order 240: $C_2^2\times A_5$ ($C_2^3$) x 2, $C_2\times S_5$ ($C_2^3$) x 2, $C_4\times A_5$ ($C_2^3$), $A_5:C_4$ ($C_2^3$), $C_2\times S_5$ ($D_4$)

Order 120: $C_2\times A_5$ ($C_2^4$), $C_2\times A_5$ ($C_2\times D_4$), $S_5$ ($C_2\times D_4$)

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

Order 16: $C_2\times D_4$ ($S_5$)

Order 8: $C_2^3$ ($C_2\times S_5$), $D_4$ ($C_2\times S_5$), $C_2\times C_4$ ($C_2\times S_5$)

Order 4: $C_2^2$ ($C_2^2\times S_5$) x 2, $C_4$ ($C_2^2\times S_5$)

Order 2: $C_2$ ($C_2^3\times S_5$), $C_2$ ($D_4\times S_5$)

Order 1: $C_1$ ($C_2\times D_4\times S_5$)

Series

Derived series

$C_2\times D_4\times S_5$

$\rhd$

$C_2\times A_5$

$\rhd$

$A_5$

Chief series

$C_2\times D_4\times S_5$

$\rhd$

$C_2\times D_4\times A_5$

$\rhd$

$C_2\times D_4$

$\rhd$

$D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times D_4\times S_5$

$\rhd$

$C_2\times A_5$

$\rhd$

$A_5$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 29 larger groups in the database.

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $70 \times 70$ rational character table. Alternatively, you may search for characters of this group with desired properties.