Abstract group 48.33: $\SL(2,3):C_2$

• Label: 48.33
• Order: \( 2^{4} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$\SL(2,3):C_2$
Order:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \)
Composition factors:	$C_2$ x 4, $C_3$
Derived length:	$3$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	8	8	8	16	48
Conjugacy classes	1	2	2	3	2	4	14
Divisions	1	2	1	2	1	1	8
Autjugacy classes	1	2	1	2	1	1	8

Dimension	1	2	3	4	8
Irr. complex chars.	6	6	2	0	0	14
Irr. rational chars.	2	2	2	1	1	8

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c \mid c^{4}=1, a^{6}=c^{2}, b^{2}=c^{2}, b^{a}=c^{3}, c^{a}=bc^{3}, c^{b}=c^{3} \rangle$
Permutation group:	Degree $16$ $\langle(1,10,2,9)(3,12,4,11)(5,14,6,13)(7,16,8,15), (3,7,5)(4,8,6)(11,15,13)(12,16,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 1 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & -1 \\ -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rr} 2 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 3 & 3 \\ 4 & 1 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{5})$
Transitive group:	16T60	24T21			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\SL(2,3)$ $\,\rtimes\,$ $C_2$	$(D_4:C_2)$ $\,\rtimes\,$ $C_3$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $A_4$	$Q_8$ . $C_6$	$C_2$ . $(C_2\times A_4)$		more information

Elements of the group are displayed as matrices in $\GL_{2}(\F_{5})$.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 37 subgroups in 15 conjugacy classes, 7 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $A_4$
Commutator:	$G' \simeq$ $Q_8$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_2$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $\SL(2,3):C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2\times A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 48: $\SL(2,3):C_2$

Order 24: $\SL(2,3)$

Order 16: $D_4:C_2$

Order 12: $C_{12}$

Order 8: $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $C_6$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 48: $\SL(2,3):C_2$

Order 24: $\SL(2,3)$

Order 16: $D_4:C_2$

Order 12: $C_{12}$

Order 8: $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $C_6$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 48: $\SL(2,3):C_2$ ($C_1$)

Order 24: $\SL(2,3)$ ($C_2$)

Order 16: $D_4:C_2$ ($C_3$)

Order 8: $Q_8$ ($C_6$)

Order 4: $C_4$ ($A_4$)

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

Order 1: $C_1$ ($\SL(2,3):C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 48: $\SL(2,3):C_2$ ($C_1$)

Order 24: $\SL(2,3)$ ($C_2$)

Order 16: $D_4:C_2$ ($C_3$)

Order 8: $Q_8$ ($C_6$)

Order 4: $C_4$ ($A_4$)

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

Order 1: $C_1$ ($\SL(2,3):C_2$)

Series

Derived series	$\SL(2,3):C_2$	$\rhd$	$Q_8$	$\rhd$	$C_2$	$\rhd$	$C_1$
Chief series	$\SL(2,3):C_2$	$\rhd$	$\SL(2,3)$	$\rhd$	$Q_8$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$\SL(2,3):C_2$	$\rhd$	$Q_8$
Upper central series	$C_1$	$\lhd$	$C_4$

Supergroups

This group is a maximal subgroup of 66 larger groups in the database.

This group is a maximal quotient of 58 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	3A1	3A-1	4A1	4A-1	4B	6A1	6A-1	12A1	12A-1	12A5	12A-5
	Size	1	1	6	4	4	1	1	6	4	4	4	4	4	4
	2 P	1A	1A	1A	3A-1	3A1	2A	2A	2A	3A1	3A-1	6A1	6A-1	6A-1	6A1
	3 P	1A	2A	2B	1A	1A	4A-1	4A1	4B	2A	2A	4A1	4A-1	4A1	4A-1
	Type
48.33.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.33.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
48.33.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$
48.33.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$
48.33.1d1	C	$1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$
48.33.1d2	C	$1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$
48.33.2a1	C	$2$	$-2$	$0$	$-1$	$-1$	$-2i$	$2i$	$0$	$1$	$1$	$-i$	$i$	$i$	$-i$
48.33.2a2	C	$2$	$-2$	$0$	$-1$	$-1$	$2i$	$-2i$	$0$	$1$	$1$	$i$	$-i$	$-i$	$i$
48.33.2b1	C	$2$	$-2$	$0$	$-{\zeta }_{12}^4$	${\zeta }_{12}^2$	$-2{\zeta }_{12}^3$	$2{\zeta }_{12}^3$	$0$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	${\zeta }_{12}$	$-{\zeta }_{12}$	$-{\zeta }_{12}^5$	${\zeta }_{12}^5$
48.33.2b2	C	$2$	$-2$	$0$	${\zeta }_{12}^2$	$-{\zeta }_{12}^4$	$2{\zeta }_{12}^3$	$-2{\zeta }_{12}^3$	$0$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$-{\zeta }_{12}^5$	${\zeta }_{12}^5$	${\zeta }_{12}$	$-{\zeta }_{12}$
48.33.2b3	C	$2$	$-2$	$0$	$-{\zeta }_{12}^4$	${\zeta }_{12}^2$	$2{\zeta }_{12}^3$	$-2{\zeta }_{12}^3$	$0$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	$-{\zeta }_{12}$	${\zeta }_{12}$	${\zeta }_{12}^5$	$-{\zeta }_{12}^5$
48.33.2b4	C	$2$	$-2$	$0$	${\zeta }_{12}^2$	$-{\zeta }_{12}^4$	$-2{\zeta }_{12}^3$	$2{\zeta }_{12}^3$	$0$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	${\zeta }_{12}^5$	$-{\zeta }_{12}^5$	$-{\zeta }_{12}$	${\zeta }_{12}$
48.33.3a	R	$3$	$3$	$-1$	$0$	$0$	$3$	$3$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$
48.33.3b	R	$3$	$3$	$1$	$0$	$0$	$-3$	$-3$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$

Rational character table

		1A	2A	2B	3A	4A	4B	6A	12A
	Size	1	1	6	8	2	6	8	16
	2 P	1A	1A	1A	3A	2A	2A	3A	6A
	3 P	1A	2A	2B	1A	4A	4B	2A	4A
48.33.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.33.1b		$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$
48.33.1c		$2$	$2$	$2$	$-1$	$2$	$2$	$-1$	$-1$
48.33.1d		$2$	$2$	$-2$	$-1$	$-2$	$2$	$-1$	$1$
48.33.2a		$4$	$-4$	$0$	$-2$	$0$	$0$	$2$	$0$
48.33.2b		$8$	$-8$	$0$	$2$	$0$	$0$	$-2$	$0$
48.33.3a		$3$	$3$	$-1$	$0$	$3$	$-1$	$0$	$0$
48.33.3b		$3$	$3$	$1$	$0$	$-3$	$-1$	$0$	$0$