Abstract group $C_6\times Q_8$

• Label: 48.46
• Order: \( 2^{4} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_6\times Q_8$
Order:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) (generators)
Outer automorphisms:	$C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \)
Composition factors:	$C_2$ x 4, $C_3$
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	3	2	12	6	24	48
Conjugacy classes	1	3	2	6	6	12	30
Divisions	1	3	1	6	3	6	20
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	4
Irr. complex chars.	24	6	0	30
Irr. rational chars.	8	10	2	20

Minimal Presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$91$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	5	5

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{12}=[a,b]=[a,c]=1, b^{2}=c^{6}, c^{b}=c^{7} \rangle$
Permutation group:	Degree $13$ $\langle(1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7), (9,10), (11,13,12), (1,4)(2,6)(3,8)(5,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrr} -1 & 1 & 1 & 0 & 0 \\ -1 & 1 & 1 & -1 & 0 \\ -1 & 0 & 0 & 1 & 0 \\ -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 \\ -1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 4 & 0 & 1 \\ 3 & 6 & 1 \\ 5 & 5 & 5 \end{array}\right), \left(\begin{array}{rrr} 2 & 1 & 6 \\ 0 & 6 & 0 \\ 5 & 4 & 2 \end{array}\right), \left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrr} 1 & 1 & 1 \\ 1 & 5 & 5 \\ 3 & 3 & 6 \end{array}\right), \left(\begin{array}{rrr} 0 & 5 & 2 \\ 0 & 6 & 0 \\ 4 & 6 & 0 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{7})$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $Q_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $C_2^3$	$C_{12}$ . $C_2^2$ (6)	$(C_2\times C_4)$ . $C_6$ (3)	$C_4$ . $(C_2\times C_6)$ (6)	all 8

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

Homology

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

Subgroups

There are 38 subgroups, all normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times Q_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_6\times Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_8$
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: $C_6\times Q_8$

Order 24: $C_3\times Q_8$ x 4, $C_2\times C_{12}$ x 3

Order 16: $C_2\times Q_8$

Order 12: $C_{12}$ x 6, $C_2\times C_6$

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 48: $C_6\times Q_8$

Order 24: $C_2\times C_{12}$, $C_3\times Q_8$

Order 16: $C_2\times Q_8$

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

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 48: $C_6\times Q_8$ ($C_1$)

Order 24: $C_3\times Q_8$ ($C_2$) x 4, $C_2\times C_{12}$ ($C_2$) x 3

Order 16: $C_2\times Q_8$ ($C_3$)

Order 12: $C_{12}$ ($C_2^2$) x 6, $C_2\times C_6$ ($C_2^2$)

Order 8: $Q_8$ ($C_6$) x 4, $C_2\times C_4$ ($C_6$) x 3

Order 6: $C_6$ ($Q_8$) x 2, $C_6$ ($C_2^3$)

Order 4: $C_4$ ($C_2\times C_6$) x 6, $C_2^2$ ($C_2\times C_6$)

Order 3: $C_3$ ($C_2\times Q_8$)

Order 2: $C_2$ ($C_3\times Q_8$) x 2, $C_2$ ($C_2^2\times C_6$)

Order 1: $C_1$ ($C_6\times Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 48: $C_6\times Q_8$ ($C_1$)

Order 24: $C_2\times C_{12}$ ($C_2$), $C_3\times Q_8$ ($C_2$)

Order 16: $C_2\times Q_8$ ($C_3$)

Order 12: $C_2\times C_6$ ($C_2^2$), $C_{12}$ ($C_2^2$)

Order 8: $Q_8$ ($C_6$), $C_2\times C_4$ ($C_6$)

Order 6: $C_6$ ($C_2^3$), $C_6$ ($Q_8$)

Order 4: $C_2^2$ ($C_2\times C_6$), $C_4$ ($C_2\times C_6$)

Order 3: $C_3$ ($C_2\times Q_8$)

Order 2: $C_2$ ($C_2^2\times C_6$), $C_2$ ($C_3\times Q_8$)

Order 1: $C_1$ ($C_6\times Q_8$)

Series

Derived series	$C_6\times Q_8$	$\rhd$	$C_2$	$\rhd$	$C_1$
Chief series	$C_6\times Q_8$	$\rhd$	$C_2\times C_{12}$	$\rhd$	$C_2\times C_6$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_6\times Q_8$	$\rhd$	$C_2$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2\times C_6$	$\lhd$	$C_6\times Q_8$

Supergroups

This group is a maximal subgroup of 87 larger groups in the database.

This group is a maximal quotient of 64 larger groups in the database.

Character theory

Complex character table

See the $30 \times 30$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.