Abstract group 16.11: $C_2\times D_4$

• Label: 16.11
• Order: \( 2^{4} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $6$
• Trans deg.: $8$
• Rank: $3$

Group information

Description:	$C_2\times D_4$
Order:	\(16\)\(\medspace = 2^{4} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \)
Composition factors:	$C_2$ x 4
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.

Group statistics

Order	1	2	4
Elements	1	11	4	16
Conjugacy classes	1	7	2	10
Divisions	1	7	2	10
Autjugacy classes	1	3	1	5

Dimension	1	2
Irr. complex chars.	8	2	10
Irr. rational chars.	8	2	10

Minimal presentations

Permutation degree:	$6$
Transitive degree:	$8$
Rank:	$3$
Inequivalent generating triples:	$21$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{4}=[a,b]=[a,c]=1, c^{b}=c^{3} \rangle$
Permutation group:	$\langle(1,2)(3,4), (2,4)(5,6), (1,3)(2,4)(5,6), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{3}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 1 & 1 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 2 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{3})$
Transitive group:	8T9	16T9			more information
Direct product:	$C_2$ $\, \times\, $ $D_4$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $C_2$ (2)	$C_4$ $\,\rtimes\,$ $C_2^2$ (2)	$C_2^2$ $\,\rtimes\,$ $C_2^2$ (4)	$(C_2\times C_4)$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $C_2^3$	$C_2^2$ . $C_2^2$			more information
Aut. group:	$\Aut(C_2\times C_8)$	$\Aut(\OD_{16})$	$\Aut(\SD_{16})$	$\Aut(C_2\times C_{12})$	all 5

Elements of the group are displayed as permutations of degree 6.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 35 subgroups in 27 conjugacy classes, 19 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$

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 16: $C_2\times D_4$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 16: $C_2\times D_4$

Order 8: $C_2^3$, $D_4$, $C_2\times C_4$

Order 4: $C_2^2$ x 3, $C_4$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16: $C_2\times D_4$ ($C_1$)

Order 8: $D_4$ ($C_2$) x 4, $C_2^3$ ($C_2$) x 2, $C_2\times C_4$ ($C_2$)

Order 4: $C_2^2$ ($C_2^2$) x 5, $C_4$ ($C_2^2$) x 2

Order 2: $C_2$ ($D_4$) x 2, $C_2$ ($C_2^3$)

Order 1: $C_1$ ($C_2\times D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16: $C_2\times D_4$ ($C_1$)

Order 8: $C_2^3$ ($C_2$), $D_4$ ($C_2$), $C_2\times C_4$ ($C_2$)

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

Order 2: $C_2$ ($C_2^3$), $C_2$ ($D_4$)

Order 1: $C_1$ ($C_2\times D_4$)

Series

Derived series	$C_2\times D_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Chief series	$C_2\times D_4$	$\rhd$	$D_4$	$\rhd$	$C_2^2$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2\times D_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2^2$	$\lhd$	$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 136 larger groups in the database.

This group is a maximal quotient of 243 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

		1A	2A	2B	2C	2D	2E	2F	2G	4A	4B
	Size	1	1	1	1	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	2C	2C
16.11.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.11.1b		$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
16.11.1c		$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
16.11.1d		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
16.11.1e		$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
16.11.1f		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
16.11.1g		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
16.11.1h		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
16.11.2a		$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
16.11.2b		$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$