Abstract group 16.14: $C_2^4$

• Label: 16.14
• Order: \( 2^{4} \)
• Exponent: \( 2 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Perm deg.: $8$
• Trans deg.: $16$
• Rank: $4$

Group information

Description:	$C_2^4$
Order:	\(16\)\(\medspace = 2^{4} \)
Exponent:	\(2\)
Automorphism group:	$A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2
Elements	1	15	16
Conjugacy classes	1	15	16
Divisions	1	15	16
Autjugacy classes	1	1	2

Dimension	1
Irr. complex chars.	16	16
Irr. rational chars.	16	16

Minimal presentations

Permutation degree:	$8$
Transitive degree:	$16$
Rank:	$4$
Inequivalent generating quadruples:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	4	4

Constructions

Presentation:	Abelian group $\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{2}=1 \rangle$
Permutation group:	$\langle(1,2), (3,4), (5,6), (7,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{2})$
Transitive group:	16T3				more information
Direct product:	$C_2$ ${}^4$
Semidirect product:	not isomorphic to a non-trivial semidirect product
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

Primary decomposition:	$C_{2}^{4}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$0$

Subgroups

There are 67 subgroups, all normal (2 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$

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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Subgroup information

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

Classes of subgroups up to conjugation

Order 16: $C_2^4$

Order 8: $C_2^3$ x 15

Order 4: $C_2^2$ x 35

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16: $C_2^4$ ($C_1$)

Order 8: $C_2^3$ ($C_2$) x 15

Order 4: $C_2^2$ ($C_2^2$) x 35

Order 2: $C_2$ ($C_2^3$) x 15

Order 1: $C_1$ ($C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16: $C_2^4$ ($C_1$)

Order 8: $C_2^3$ ($C_2$)

Order 4: $C_2^2$ ($C_2^2$)

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

Order 1: $C_1$ ($C_2^4$)

Series

Derived series	$C_2^4$	$\rhd$	$C_1$
Chief series	$C_2^4$	$\rhd$	$C_2^3$	$\rhd$	$C_2^2$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2^4$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2^4$

Supergroups

This group is a maximal subgroup of 68 larger groups in the database.

This group is a maximal quotient of 82 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	2H	2I	2J	2K	2L	2M	2N	2O
	Size	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A
16.14.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.14.1b		$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$
16.14.1c		$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$
16.14.1d		$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$
16.14.1e		$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
16.14.1f		$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
16.14.1g		$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$
16.14.1h		$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
16.14.1i		$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
16.14.1j		$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
16.14.1k		$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
16.14.1l		$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
16.14.1m		$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$
16.14.1n		$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$
16.14.1o		$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$
16.14.1p		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$