Abstract group $C_3.A_6$

• Label: 1080.260
• Order: \( 2^{3} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{5} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $18$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_3.A_6$
Order:	\(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \)
Outer automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:	$C_3$, $A_6$
Derived length:	$0$

This group is nonabelian and quasisimple (hence nonsolvable and perfect).

Group statistics

Order	1	2	3	4	5	6	12	15
Elements	1	45	242	90	144	90	180	288	1080
Conjugacy classes	1	1	4	1	2	2	2	4	17
Divisions	1	1	3	1	1	1	1	1	10
Autjugacy classes	1	1	2	1	1	1	1	1	9

Dimension	1	3	5	6	8	9	10	12	15	16	18	30
Irr. complex chars.	1	4	2	2	2	3	1	0	2	0	0	0	17
Irr. rational chars.	1	0	2	0	0	1	1	2	0	1	1	1	10

Minimal Presentations

Permutation degree:	$18$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$477$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	12
Arbitrary	3	6	12

Constructions

Permutation group:	Degree $18$ $\langle(1,2,4)(3,7,14)(5,10,11)(6,12,9)(8,15,16), (1,3,6)(2,5,9)(4,8,12)(7,14,13)(10,11,17)(15,16,18)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} z_{2} & z_{2} & z_{2} \\ 0 & z_{2} + 1 & z_{2} + 1 \\ z_{2} & z_{2} + 1 & z_{2} \end{array}\right), \left(\begin{array}{rrr} 1 & 1 & z_{2} \\ z_{2} & z_{2} & z_{2} \\ z_{2} + 1 & z_{2} & 0 \end{array}\right), \left(\begin{array}{rrr} z_{2} + 1 & 0 & 0 \\ 0 & z_{2} + 1 & 0 \\ 0 & 0 & z_{2} + 1 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{4})$
Transitive group:	18T262	45T150			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $A_6$				more information

Elements of the group are displayed as matrices in $\GL_{3}(\F_{4})$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_{2}$
Commutator length:	$2$

Subgroups

There are 1111 subgroups in 42 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $A_6$
Commutator:	$G' \simeq$ $C_3.A_6$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $A_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3$	$G/\operatorname{Fit} \simeq$ $A_6$
Radical:	$R \simeq$ $C_3$	$G/R \simeq$ $A_6$
Socle:	$\operatorname{soc} \simeq$ $C_3$	$G/\operatorname{soc} \simeq$ $A_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1080: $C_3.A_6$

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

Order 108: $\He_3:C_4$

Order 72: $C_3\times S_4$ x 2

Order 60: $A_5$ x 2

Order 54: $C_3^2:S_3$

Order 36: $C_3\times A_4$ x 2

Order 30: $C_3\times D_5$

Order 27: $\He_3$

Order 24: $S_4$ x 2, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 2

Order 15: $C_{15}$

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

Order 10: $D_5$

Order 9: $C_3^2$ x 2

Order 8: $D_4$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1080: $C_3.A_6$

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

Order 108: $\He_3:C_4$

Order 72: $C_3\times S_4$

Order 60: $A_5$

Order 54: $C_3^2:S_3$

Order 36: $C_3\times A_4$

Order 30: $C_3\times D_5$

Order 27: $\He_3$

Order 24: $S_4$, $C_3\times D_4$

Order 18: $C_3\times S_3$

Order 15: $C_{15}$

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

Order 10: $D_5$

Order 9: $C_3^2$

Order 8: $D_4$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1080: $C_3.A_6$ ($C_1$)

Order 3: $C_3$ ($A_6$)

Order 1: $C_1$ ($C_3.A_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1080: $C_3.A_6$ ($C_1$)

Order 3: $C_3$ ($A_6$)

Order 1: $C_1$ ($C_3.A_6$)

Series

Derived series	$C_3.A_6$
Chief series	$C_3.A_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_3.A_6$
Upper central series	$C_1$	$\lhd$	$C_3$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

See the $17 \times 17$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	3B	3C	4A	5A	6A	12A	15A
	Size	1	45	2	120	120	90	144	90	180	288
	2 P	1A	1A	3A	3B	3C	2A	5A	3A	6A	15A
	3 P	1A	2A	1A	1A	1A	4A	5A	2A	4A	5A
	5 P	1A	2A	3A	3B	3C	4A	1A	6A	12A	3A
1080.260.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1080.260.3a		$12$	$-4$	$-6$	$0$	$0$	$4$	$2$	$2$	$-2$	$-1$
1080.260.5a		$5$	$1$	$5$	$-1$	$2$	$-1$	$0$	$1$	$-1$	$0$
1080.260.5b		$5$	$1$	$5$	$2$	$-1$	$-1$	$0$	$1$	$-1$	$0$
1080.260.6a		$12$	$4$	$-6$	$0$	$0$	$0$	$2$	$-2$	$0$	$-1$
1080.260.8a		$16$	$0$	$16$	$-2$	$-2$	$0$	$1$	$0$	$0$	$1$
1080.260.9a		$9$	$1$	$9$	$0$	$0$	$1$	$-1$	$1$	$1$	$-1$
1080.260.9b		$18$	$2$	$-9$	$0$	$0$	$2$	$-2$	$-1$	$-1$	$1$
1080.260.10a		$10$	$-2$	$10$	$1$	$1$	$0$	$0$	$-2$	$0$	$0$
1080.260.15a		$30$	$-2$	$-15$	$0$	$0$	$-2$	$0$	$1$	$1$	$0$