Abstract group 480.1192: $C_2\times A_4\times F_5$

• Label: 480.1192
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $11$
• Trans deg.: $30$
• Rank: $2$

Group information

Description:	$C_2\times A_4\times F_5$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2\times F_5\times S_4$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
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	47	8	80	4	88	28	160	32	32	480
Conjugacy classes	1	7	2	8	1	6	3	8	2	2	40
Divisions	1	7	1	4	1	3	3	2	1	1	24
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	8	6	0	0	2	40
Irr. rational chars.	4	6	4	4	2	2	2	24

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$30$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	7	7	7

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{12}=c^{2}=d^{10}=[a,b]=[a,c]=[a,d]=[c,d]=1, c^{b}=d^{5}, d^{b}=cd^{3} \rangle$
Permutation group:	Degree $11$ $\langle(1,2)(3,4)(5,6)(8,9,10,11), (1,2)(3,4)(5,6), (1,3,5)(2,4,6), (8,10)(9,11), (3,4)(5,6), (1,2)(3,4), (7,8,9,11,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 11 & 10 \\ 10 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 10 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 10 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Transitive group:	30T107	40T426			more information
Direct product:	$C_2$ $\, \times\, $ $A_4$ $\, \times\, $ $F_5$
Semidirect product:	$(D_5\times A_4)$ $\,\rtimes\,$ $C_4$	$D_5$ $\,\rtimes\,$ $(C_4\times A_4)$	$(C_{10}\times A_4)$ $\,\rtimes\,$ $C_4$	$C_{10}$ $\,\rtimes\,$ $(C_4\times A_4)$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(A_4\times D_{10})$ . $C_2$	$D_{10}$ . $(C_2\times A_4)$	$(D_5\times A_4)$ . $C_2^2$	$D_5$ . $(C_2^2\times A_4)$	all 6

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 844 subgroups in 132 conjugacy classes, 30 normal (24 characteristic).

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

Special subgroups

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 480: $C_2\times A_4\times F_5$

Order 240: $A_4\times F_5$ x 2, $A_4\times D_{10}$

Order 160: $C_2^3\times F_5$

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

Order 96: $C_2^3:C_{12}$

Order 80: $C_2^2\times F_5$ x 6, $C_2^2\times D_{10}$

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

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

Order 40: $C_2\times F_5$ x 10, $C_2\times D_{10}$ x 6, $C_2^2\times C_{10}$

Order 32: $C_2^3\times C_4$

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 10, $F_5$ x 4, $C_2\times C_{10}$ x 3

Order 16: $C_2^2\times C_4$ x 6, $C_2^4$

Order 15: $C_{15}$

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

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

Order 8: $C_2\times C_4$ x 10, $C_2^3$ x 7

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 13, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_2\times A_4\times F_5$

Order 240: $A_4\times D_{10}$, $A_4\times F_5$

Order 160: $C_2^3\times F_5$

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

Order 96: $C_2^3:C_{12}$

Order 80: $C_2^2\times F_5$ x 4, $C_2^2\times D_{10}$

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

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

Order 40: $C_2\times D_{10}$ x 6, $C_2\times F_5$ x 6, $C_2^2\times C_{10}$

Order 32: $C_2^3\times C_4$

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 9, $C_2\times C_{10}$ x 3, $F_5$ x 2

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

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 7, $C_2\times C_4$ x 6

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_2\times A_4\times F_5$ ($C_1$)

Order 240: $A_4\times F_5$ ($C_2$) x 2, $A_4\times D_{10}$ ($C_2$)

Order 160: $C_2^3\times F_5$ ($C_3$)

Order 120: $C_{10}\times A_4$ ($C_4$), $D_5\times A_4$ ($C_2^2$), $D_5\times A_4$ ($C_4$)

Order 80: $C_2^2\times F_5$ ($C_6$) x 2, $C_2^2\times D_{10}$ ($C_6$)

Order 60: $C_5\times A_4$ ($C_2\times C_4$)

Order 40: $C_2^2\times C_{10}$ ($C_{12}$), $C_2\times D_{10}$ ($C_2\times C_6$), $C_2\times D_{10}$ ($C_{12}$), $C_2\times F_5$ ($A_4$)

Order 24: $C_2\times A_4$ ($F_5$)

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

Order 12: $A_4$ ($C_2\times F_5$)

Order 10: $C_{10}$ ($C_4\times A_4$), $D_5$ ($C_2^2\times A_4$), $D_5$ ($C_4\times A_4$)

Order 8: $C_2^3$ ($C_3\times F_5$)

Order 5: $C_5$ ($C_2^3:C_{12}$)

Order 4: $C_2^2$ ($C_6\times F_5$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_2\times A_4\times F_5$ ($C_1$)

Order 240: $A_4\times D_{10}$ ($C_2$), $A_4\times F_5$ ($C_2$)

Order 160: $C_2^3\times F_5$ ($C_3$)

Order 120: $C_{10}\times A_4$ ($C_4$), $D_5\times A_4$ ($C_2^2$), $D_5\times A_4$ ($C_4$)

Order 80: $C_2^2\times D_{10}$ ($C_6$), $C_2^2\times F_5$ ($C_6$)

Order 60: $C_5\times A_4$ ($C_2\times C_4$)

Order 40: $C_2^2\times C_{10}$ ($C_{12}$), $C_2\times D_{10}$ ($C_2\times C_6$), $C_2\times D_{10}$ ($C_{12}$), $C_2\times F_5$ ($A_4$)

Order 24: $C_2\times A_4$ ($F_5$)

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

Order 12: $A_4$ ($C_2\times F_5$)

Order 10: $C_{10}$ ($C_4\times A_4$), $D_5$ ($C_2^2\times A_4$), $D_5$ ($C_4\times A_4$)

Order 8: $C_2^3$ ($C_3\times F_5$)

Order 5: $C_5$ ($C_2^3:C_{12}$)

Order 4: $C_2^2$ ($C_6\times F_5$)

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

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

Series

Derived series

$C_2\times A_4\times F_5$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_1$

Chief series

$C_2\times A_4\times F_5$

$\rhd$

$A_4\times D_{10}$

$\rhd$

$D_5\times A_4$

$\rhd$

$C_5\times A_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2\times A_4\times F_5$

$\rhd$

$C_2\times C_{10}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 31 larger groups in the database.

This group is a maximal quotient of 35 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.