Abstract group 1920.240406: $C_2^3:A_4\times F_5$

• Label: 1920.240406
• Order: \( 2^{7} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \cdot 5 \)
• Perm deg.: $15$
• Trans deg.: $60$
• Rank: $3$

Group information

Description:	$C_2^3:A_4\times F_5$
Order:	\(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$F_5\times C_2^4.(C_6\times A_5).C_2$, of order \(230400\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{2} \)
Composition factors:	$C_2$ x 7, $C_3$, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	191	32	320	4	352	124	640	128	128	1920
Conjugacy classes	1	23	2	24	1	6	11	8	2	2	80
Divisions	1	23	1	12	1	3	11	2	1	1	56
Autjugacy classes	1	7	1	4	1	3	3	2	1	1	24

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	24	0	40	6	0	0	10	80
Irr. rational chars.	4	6	20	4	10	2	10	56

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$60$
Rank:	$3$
Inequivalent generating triples:	$13104$

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

Presentation:	${\langle a, b, c, d, e, f \mid a^{12}=b^{10}=c^{2}=d^{2}=e^{2}=f^{2}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(10,11,12,13)(14,15), (14,15), (2,4,3)(6,8,7), (10,12)(11,13), (5,6)(7,8), (5,7)(6,8), (1,2)(3,4)(5,8)(6,7), (1,3)(2,4)(5,6)(7,8), (9,10,11,13,12)\rangle$
Direct product:	$C_2$ $\, \times\, $ $F_5$ $\, \times\, $ $(C_2^2:A_4)$
Semidirect product:	$(C_2^3\times F_5)$ $\,\rtimes\,$ $A_4$	$C_2^3$ $\,\rtimes\,$ $(A_4\times F_5)$	$(F_5\times C_2^5)$ $\,\rtimes\,$ $C_3$	$(C_2^4\times F_5)$ $\,\rtimes\,$ $C_6$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $(C_2^3:A_4)$	$(C_2^4\times D_{10})$ . $C_6$	$D_5$ . $(C_2^2\wr C_3)$	$(C_2^2:A_4\times D_{10})$ . $C_2$	all 8

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 31640 subgroups in 3704 conjugacy classes, 80 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2:A_4\times F_5$
Commutator:	$G' \simeq$ $C_2^3\times C_{10}$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^3:A_4\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_{12}$
Radical:	$R \simeq$ $C_2^3:A_4\times F_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5\times C_4$
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: $C_2^3:A_4\times F_5$

Order 960: $C_2^2:A_4\times F_5$ x 2, $C_2^2:A_4\times D_{10}$

Order 640: $F_5\times C_2^5$

Order 480: $C_2\times A_4\times F_5$ x 5, $D_5\times C_2^2:A_4$ x 2, $C_2^4:C_{30}$

Order 384: $C_2^5:C_{12}$

Order 320: $C_2^4\times F_5$ x 22, $C_2^4\times D_{10}$

Order 240: $A_4\times F_5$ x 10, $A_4\times D_{10}$ x 5, $C_2^4:C_{15}$

Order 192: $C_2^4:C_{12}$ x 2, $C_2^2\wr C_3$

Order 160: $C_2^3\times F_5$ x 210, $C_2^3\times D_{10}$ x 22, $C_2^4\times C_{10}$

Order 128: $C_2^5\times C_4$

Order 120: $D_5\times A_4$ x 10, $C_{10}\times A_4$ x 5, $C_6\times F_5$

Order 96: $C_2^3:C_{12}$ x 5, $C_2^3:A_4$ x 3

Order 80: $C_2^2\times F_5$ x 420, $C_2^2\times D_{10}$ x 210, $C_2^3\times C_{10}$ x 11

Order 64: $C_2^4\times C_4$ x 22, $C_2^6$

Order 60: $C_5\times A_4$ x 5, $C_3\times F_5$ x 2, $C_3\times D_{10}$

Order 48: $C_4\times A_4$ x 10, $C_2^2\times A_4$ x 5, $C_2^2:A_4$

Order 40: $C_2\times D_{10}$ x 420, $C_2\times F_5$ x 166, $C_2^2\times C_{10}$ x 55

Order 32: $C_2^3\times C_4$ x 210, $C_2^5$ x 23

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

Order 24: $C_2\times A_4$ x 15, $C_2\times C_{12}$

Order 20: $D_{10}$ x 166, $C_2\times C_{10}$ x 55, $F_5$ x 12

Order 16: $C_2^2\times C_4$ x 420, $C_2^4$ x 221

Order 15: $C_{15}$

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

Order 10: $D_5$ x 12, $C_{10}$ x 11

Order 8: $C_2^3$ x 475, $C_2\times C_4$ x 166

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 221, $C_4$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1920: $C_2^3:A_4\times F_5$

Order 960: $C_2^2:A_4\times D_{10}$, $C_2^2:A_4\times F_5$

Order 640: $F_5\times C_2^5$

Order 480: $D_5\times C_2^2:A_4$ x 2, $C_2^4:C_{30}$, $C_2\times A_4\times F_5$

Order 384: $C_2^5:C_{12}$

Order 320: $C_2^4\times F_5$ x 4, $C_2^4\times D_{10}$

Order 240: $C_2^4:C_{15}$, $A_4\times D_{10}$, $A_4\times F_5$

Order 192: $C_2^2\wr C_3$, $C_2^4:C_{12}$

Order 160: $C_2^3\times F_5$ x 10, $C_2^3\times D_{10}$ x 6, $C_2^4\times C_{10}$

Order 128: $C_2^5\times C_4$

Order 120: $D_5\times A_4$ x 2, $C_{10}\times A_4$, $C_6\times F_5$

Order 96: $C_2^3:A_4$ x 3, $C_2^3:C_{12}$

Order 80: $C_2^2\times D_{10}$ x 14, $C_2^2\times F_5$ x 12, $C_2^3\times C_{10}$ x 3

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

Order 60: $C_5\times A_4$, $C_3\times F_5$, $C_3\times D_{10}$

Order 48: $C_2^2:A_4$, $C_2^2\times A_4$, $C_4\times A_4$

Order 40: $C_2\times D_{10}$ x 19, $C_2\times F_5$ x 8, $C_2^2\times C_{10}$ x 5

Order 32: $C_2^3\times C_4$ x 10, $C_2^5$ x 7

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

Order 24: $C_2\times A_4$ x 3, $C_2\times C_{12}$

Order 20: $D_{10}$ x 12, $C_2\times C_{10}$ x 5, $F_5$ x 2

Order 16: $C_2^4$ x 17, $C_2^2\times C_4$ x 12

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $A_4$, $C_{12}$

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

Order 8: $C_2^3$ x 24, $C_2\times C_4$ x 8

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 17, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1920: $C_2^3:A_4\times F_5$ ($C_1$)

Order 960: $C_2^2:A_4\times F_5$ ($C_2$) x 2, $C_2^2:A_4\times D_{10}$ ($C_2$)

Order 640: $F_5\times C_2^5$ ($C_3$)

Order 480: $C_2^4:C_{30}$ ($C_4$), $D_5\times C_2^2:A_4$ ($C_2^2$), $D_5\times C_2^2:A_4$ ($C_4$)

Order 320: $C_2^4\times F_5$ ($C_6$) x 2, $C_2^4\times D_{10}$ ($C_6$)

Order 240: $C_2^4:C_{15}$ ($C_2\times C_4$)

Order 160: $C_2^3\times F_5$ ($A_4$) x 5, $C_2^4\times C_{10}$ ($C_{12}$), $C_2^3\times D_{10}$ ($C_2\times C_6$), $C_2^3\times D_{10}$ ($C_{12}$)

Order 96: $C_2^3:A_4$ ($F_5$)

Order 80: $C_2^2\times F_5$ ($C_2\times A_4$) x 10, $C_2^2\times D_{10}$ ($C_2\times A_4$) x 5, $C_2^3\times C_{10}$ ($C_2\times C_{12}$)

Order 48: $C_2^2:A_4$ ($C_2\times F_5$)

Order 40: $C_2^2\times C_{10}$ ($C_4\times A_4$) x 5, $C_2\times D_{10}$ ($C_2^2\times A_4$) x 5, $C_2\times D_{10}$ ($C_4\times A_4$) x 5, $C_2\times F_5$ ($C_2^2:A_4$)

Order 32: $C_2^5$ ($C_3\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_2^3:C_{12}$) x 5, $F_5$ ($C_2^3:A_4$) x 2, $D_{10}$ ($C_2^3:A_4$)

Order 16: $C_2^4$ ($C_6\times F_5$)

Order 10: $C_{10}$ ($C_2^4:C_{12}$), $D_5$ ($C_2^2\wr C_3$), $D_5$ ($C_2^4:C_{12}$)

Order 8: $C_2^3$ ($A_4\times F_5$) x 5

Order 5: $C_5$ ($C_2^5:C_{12}$)

Order 4: $C_2^2$ ($C_2\times A_4\times F_5$) x 5

Order 2: $C_2$ ($C_2^2:A_4\times F_5$)

Order 1: $C_1$ ($C_2^3:A_4\times F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1920: $C_2^3:A_4\times F_5$ ($C_1$)

Order 960: $C_2^2:A_4\times D_{10}$ ($C_2$), $C_2^2:A_4\times F_5$ ($C_2$)

Order 640: $F_5\times C_2^5$ ($C_3$)

Order 480: $C_2^4:C_{30}$ ($C_4$), $D_5\times C_2^2:A_4$ ($C_2^2$), $D_5\times C_2^2:A_4$ ($C_4$)

Order 320: $C_2^4\times D_{10}$ ($C_6$), $C_2^4\times F_5$ ($C_6$)

Order 240: $C_2^4:C_{15}$ ($C_2\times C_4$)

Order 160: $C_2^4\times C_{10}$ ($C_{12}$), $C_2^3\times D_{10}$ ($C_2\times C_6$), $C_2^3\times D_{10}$ ($C_{12}$), $C_2^3\times F_5$ ($A_4$)

Order 96: $C_2^3:A_4$ ($F_5$)

Order 80: $C_2^3\times C_{10}$ ($C_2\times C_{12}$), $C_2^2\times D_{10}$ ($C_2\times A_4$), $C_2^2\times F_5$ ($C_2\times A_4$)

Order 48: $C_2^2:A_4$ ($C_2\times F_5$)

Order 40: $C_2^2\times C_{10}$ ($C_4\times A_4$), $C_2\times D_{10}$ ($C_2^2\times A_4$), $C_2\times D_{10}$ ($C_4\times A_4$), $C_2\times F_5$ ($C_2^2:A_4$)

Order 32: $C_2^5$ ($C_3\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_2^3:C_{12}$), $D_{10}$ ($C_2^3:A_4$), $F_5$ ($C_2^3:A_4$)

Order 16: $C_2^4$ ($C_6\times F_5$)

Order 10: $C_{10}$ ($C_2^4:C_{12}$), $D_5$ ($C_2^2\wr C_3$), $D_5$ ($C_2^4:C_{12}$)

Order 8: $C_2^3$ ($A_4\times F_5$)

Order 5: $C_5$ ($C_2^5:C_{12}$)

Order 4: $C_2^2$ ($C_2\times A_4\times F_5$)

Order 2: $C_2$ ($C_2^2:A_4\times F_5$)

Order 1: $C_1$ ($C_2^3:A_4\times F_5$)

Series

Derived series

$C_2^3:A_4\times F_5$

$\rhd$

$C_2^3\times C_{10}$

$\rhd$

$C_1$

Chief series

$C_2^3:A_4\times F_5$

$\rhd$

$C_2^2:A_4\times D_{10}$

$\rhd$

$D_5\times C_2^2:A_4$

$\rhd$

$C_2^4:C_{15}$

$\rhd$

$C_2^3\times C_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2^3:A_4\times F_5$

$\rhd$

$C_2^3\times C_{10}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 0 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 $56 \times 56$ rational character table.