Abstract group 1518.1: $C_3\times F_{23}$

• Label: 1518.1
• Order: \( 2 \cdot 3 \cdot 11 \cdot 23 \)
• Exponent: \( 2 \cdot 3 \cdot 11 \cdot 23 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \cdot 11 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 11 \cdot 23 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $26$
• Trans deg.: $69$
• Rank: $2$

Group information

Description:	$C_3\times F_{23}$
Order:	\(1518\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 23 \)
Exponent:	\(1518\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 23 \)
Automorphism group:	$C_2\times F_{23}$, of order \(1012\)\(\medspace = 2^{2} \cdot 11 \cdot 23 \)
Composition factors:	$C_2$, $C_3$, $C_{11}$, $C_{23}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	11	22	23	33	66	69
Elements	1	23	2	46	230	230	22	460	460	44	1518
Conjugacy classes	1	1	2	2	10	10	1	20	20	2	69
Divisions	1	1	1	1	1	1	1	1	1	1	10
Autjugacy classes	1	1	1	1	10	10	1	10	10	1	46

Dimension	1	2	10	20	22	44
Irr. complex chars.	66	0	0	0	3	0	69
Irr. rational chars.	2	2	2	2	1	1	10

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$69$
Rank:	$2$
Inequivalent generating pairs:	$1440$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	22	44	44
Arbitrary	22	24	24

Constructions

Presentation:	$\langle a, b \mid a^{22}=b^{69}=1, b^{a}=b^{43} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,6)(5,9)(7,11)(8,13)(10,16)(12,20)(14,21)(15,17)(18,23)(19,22), (24,25,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{16} & 0 \\ \alpha^{99} & \alpha^{448} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{209} & \alpha^{290} \\ \alpha^{144} & \alpha^{73} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{264} & 0 \\ \alpha^{107} & 1 \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{176} & 0 \\ 0 & \alpha^{176} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{529}) = \GL_{2}(\F_{23}[\alpha]/(\alpha^{2} + 21 \alpha + 5))$
Direct product:	$C_3$ $\, \times\, $ $F_{23}$
Semidirect product:	$D_{23}$ $\,\rtimes\,$ $C_{33}$	$C_{69}$ $\,\rtimes\,$ $C_{22}$	$C_{23}$ $\,\rtimes\,$ $C_{66}$	$(C_3\times D_{23})$ $\,\rtimes\,$ $C_{11}$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{66} \simeq C_{2} \times C_{3} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 148 subgroups in 16 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $F_{23}$
Commutator:	$G' \simeq$ $C_{23}$	$G/G' \simeq$ $C_{66}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3\times F_{23}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{69}$	$G/\operatorname{Fit} \simeq$ $C_{22}$
Radical:	$R \simeq$ $C_3\times F_{23}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{69}$	$G/\operatorname{soc} \simeq$ $C_{22}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 1518: $C_3\times F_{23}$

Order 759: $C_{23}:C_{33}$

Order 506: $F_{23}$

Order 253: $C_{23}:C_{11}$

Order 138: $C_3\times D_{23}$

Order 69: $C_{69}$

Order 66: $C_{66}$

Order 46: $D_{23}$

Order 33: $C_{33}$

Order 23: $C_{23}$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1518: $C_3\times F_{23}$

Order 759: $C_{23}:C_{33}$

Order 506: $F_{23}$

Order 253: $C_{23}:C_{11}$

Order 138: $C_3\times D_{23}$

Order 69: $C_{69}$

Order 66: $C_{66}$

Order 46: $D_{23}$

Order 33: $C_{33}$

Order 23: $C_{23}$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1518: $C_3\times F_{23}$ ($C_1$)

Order 759: $C_{23}:C_{33}$ ($C_2$)

Order 506: $F_{23}$ ($C_3$)

Order 253: $C_{23}:C_{11}$ ($C_6$)

Order 138: $C_3\times D_{23}$ ($C_{11}$)

Order 69: $C_{69}$ ($C_{22}$)

Order 46: $D_{23}$ ($C_{33}$)

Order 23: $C_{23}$ ($C_{66}$)

Order 3: $C_3$ ($F_{23}$)

Order 1: $C_1$ ($C_3\times F_{23}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1518: $C_3\times F_{23}$ ($C_1$)

Order 759: $C_{23}:C_{33}$ ($C_2$)

Order 506: $F_{23}$ ($C_3$)

Order 253: $C_{23}:C_{11}$ ($C_6$)

Order 138: $C_3\times D_{23}$ ($C_{11}$)

Order 69: $C_{69}$ ($C_{22}$)

Order 46: $D_{23}$ ($C_{33}$)

Order 23: $C_{23}$ ($C_{66}$)

Order 3: $C_3$ ($F_{23}$)

Order 1: $C_1$ ($C_3\times F_{23}$)

Series

Derived series	$C_3\times F_{23}$	$\rhd$	$C_{23}$	$\rhd$	$C_1$
Chief series	$C_3\times F_{23}$	$\rhd$	$C_{23}:C_{33}$	$\rhd$	$C_{69}$	$\rhd$	$C_{23}$	$\rhd$	$C_1$
Lower central series	$C_3\times F_{23}$	$\rhd$	$C_{23}$
Upper central series	$C_1$	$\lhd$	$C_3$

Supergroups

This group is a maximal subgroup of 1 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 $69 \times 69$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	6A	11A	22A	23A	33A	66A	69A
	Size	1	23	2	46	230	230	22	460	460	44
	2 P	1A	1A	3A	3A	11A	11A	23A	33A	33A	69A
	3 P	1A	2A	1A	2A	11A	22A	23A	11A	22A	23A
	11 P	1A	2A	3A	6A	11A	22A	23A	33A	66A	69A
	23 P	1A	2A	3A	6A	1A	2A	23A	3A	6A	69A
1518.1.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1518.1.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$
1518.1.1c		$2$	$2$	$-1$	$-1$	$2$	$2$	$2$	$-1$	$-1$	$-1$
1518.1.1d		$2$	$-2$	$-1$	$1$	$2$	$-2$	$2$	$-1$	$1$	$-1$
1518.1.1e		$10$	$10$	$10$	$10$	$-1$	$-1$	$10$	$-1$	$-1$	$10$
1518.1.1f		$10$	$-10$	$10$	$-10$	$-1$	$1$	$10$	$-1$	$1$	$10$
1518.1.1g		$20$	$20$	$-10$	$-10$	$-2$	$-2$	$20$	$1$	$1$	$-10$
1518.1.1h		$20$	$-20$	$-10$	$10$	$-2$	$2$	$20$	$1$	$-1$	$-10$
1518.1.22a		$22$	$0$	$22$	$0$	$0$	$0$	$-1$	$0$	$0$	$-1$
1518.1.22b		$44$	$0$	$-22$	$0$	$0$	$0$	$-2$	$0$	$0$	$1$