Abstract group $S_3\times C_{23}$

• Label: 138.1
• Order: \( 2 \cdot 3 \cdot 23 \)
• Exponent: \( 2 \cdot 3 \cdot 23 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 23 \)
• $\card{Z(G)}$: \( 23 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{2} \cdot 3 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 11 \)
• Perm deg.: $26$
• Trans deg.: $69$
• Rank: $2$

Group information

Description:	$S_3\times C_{23}$
Order:	\(138\)\(\medspace = 2 \cdot 3 \cdot 23 \)
Exponent:	\(138\)\(\medspace = 2 \cdot 3 \cdot 23 \)
Automorphism group:	$S_3\times C_{22}$, of order \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) (generators)
Outer automorphisms:	$C_{22}$, of order \(22\)\(\medspace = 2 \cdot 11 \)
Composition factors:	$C_2$, $C_3$, $C_{23}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	23	46	69
Elements	1	3	2	22	66	44	138
Conjugacy classes	1	1	1	22	22	22	69
Divisions	1	1	1	1	1	1	6
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	22	44
Irr. complex chars.	46	23	0	0	69
Irr. rational chars.	2	1	2	1	6

Minimal Presentations

Permutation degree:	$26$
Transitive degree:	$69$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	44
Arbitrary	2	4	24

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{69}=1, b^{a}=b^{47} \rangle$
Permutation group:	Degree $26$ $\langle(25,26), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (24,25,26)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 13 & 12 \\ 40 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 46 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{47})$
Direct product:	$C_{23}$ $\, \times\, $ $S_3$
Semidirect product:	$C_{69}$ $\,\rtimes\,$ $C_2$	$C_3$ $\,\rtimes\,$ $C_{46}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{46} \simeq C_{2} \times C_{23}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 12 subgroups in 8 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{23}$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_{46}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times C_{23}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{69}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $S_3\times C_{23}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{69}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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

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Classes of subgroups up to conjugation

Order 138: $S_3\times C_{23}$

Order 69: $C_{69}$

Order 46: $C_{46}$

Order 23: $C_{23}$

Order 6: $S_3$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 138: $S_3\times C_{23}$

Order 69: $C_{69}$

Order 46: $C_{46}$

Order 23: $C_{23}$

Order 6: $S_3$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 138: $S_3\times C_{23}$ ($C_1$)

Order 69: $C_{69}$ ($C_2$)

Order 23: $C_{23}$ ($S_3$)

Order 6: $S_3$ ($C_{23}$)

Order 3: $C_3$ ($C_{46}$)

Order 1: $C_1$ ($S_3\times C_{23}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 138: $S_3\times C_{23}$ ($C_1$)

Order 69: $C_{69}$ ($C_2$)

Order 23: $C_{23}$ ($S_3$)

Order 6: $S_3$ ($C_{23}$)

Order 3: $C_3$ ($C_{46}$)

Order 1: $C_1$ ($S_3\times C_{23}$)

Series

Derived series	$S_3\times C_{23}$	$\rhd$	$C_3$	$\rhd$	$C_1$
Chief series	$S_3\times C_{23}$	$\rhd$	$C_{69}$	$\rhd$	$C_{23}$	$\rhd$	$C_1$
Lower central series	$S_3\times C_{23}$	$\rhd$	$C_3$
Upper central series	$C_1$	$\lhd$	$C_{23}$

Supergroups

This group is a maximal subgroup of 17 larger groups in the database.

This group is a maximal quotient of 15 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	23A	46A	69A
	Size	1	3	2	22	66	44
	2 P	1A	1A	3A	23A	23A	69A
	3 P	1A	2A	1A	23A	46A	23A
	23 P	1A	2A	3A	23A	46A	69A
138.1.1a		$1$	$1$	$1$	$1$	$1$	$1$
138.1.1b		$1$	$-1$	$1$	$1$	$-1$	$1$
138.1.1c		$22$	$22$	$22$	$-1$	$-1$	$-1$
138.1.1d		$22$	$-22$	$22$	$-1$	$1$	$-1$
138.1.2a		$2$	$0$	$-1$	$2$	$0$	$-1$
138.1.2b		$44$	$0$	$-22$	$-2$	$0$	$1$