Abstract group 32.18: $D_{16}$

• Label: 32.18
• Order: \( 2^{5} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$D_{16}$
Order:	\(32\)\(\medspace = 2^{5} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$D_{16}:C_4$, of order \(128\)\(\medspace = 2^{7} \)
Composition factors:	$C_2$ x 5
Nilpotency class:	$4$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16
Elements	1	17	2	4	8	32
Conjugacy classes	1	3	1	2	4	11
Divisions	1	3	1	1	1	7
Autjugacy classes	1	2	1	1	1	6

Dimension	1	2	4	8
Irr. complex chars.	4	7	0	0	11
Irr. rational chars.	4	1	1	1	7

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	8
Arbitrary	2	2	8

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{16}=1, b^{a}=b^{15} \rangle$
Permutation group:	Degree $16$ $\langle(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (3,4)(5,7)(6,8)(9,15)(10,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 6 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{17})$
Transitive group:	16T56	32T31			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_8$ $\,\rtimes\,$ $C_2$ (2)	$C_{16}$ $\,\rtimes\,$ $C_2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $D_4$	$C_2$ . $D_8$	$C_8$ . $C_2^2$		more information

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

Homology

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

Subgroups

There are 36 subgroups in 14 conjugacy classes, 8 normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_8$
Commutator:	$G' \simeq$ $C_8$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $D_{16}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_{16}$

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 32: $D_{16}$

Order 16: $D_8$ x 2, $C_{16}$

Order 8: $D_4$ x 2, $C_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 32: $D_{16}$

Order 16: $D_8$, $C_{16}$

Order 8: $D_4$, $C_8$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 32: $D_{16}$ ($C_1$)

Order 16: $D_8$ ($C_2$) x 2, $C_{16}$ ($C_2$)

Order 8: $C_8$ ($C_2^2$)

Order 4: $C_4$ ($D_4$)

Order 2: $C_2$ ($D_8$)

Order 1: $C_1$ ($D_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 32: $D_{16}$ ($C_1$)

Order 16: $D_8$ ($C_2$), $C_{16}$ ($C_2$)

Order 8: $C_8$ ($C_2^2$)

Order 4: $C_4$ ($D_4$)

Order 2: $C_2$ ($D_8$)

Order 1: $C_1$ ($D_{16}$)

Series

Derived series

$D_{16}$

$\rhd$

$C_8$

$\rhd$

$C_1$

Chief series

$D_{16}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

$\lhd$

$D_{16}$

Supergroups

This group is a maximal subgroup of 67 larger groups in the database.

This group is a maximal quotient of 67 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	4A	8A1	8A3	16A1	16A3	16A5	16A7
	Size	1	1	8	8	2	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	2A	4A	4A	8A1	8A3	8A3	8A1
	Type
32.18.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.18.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.18.1c	R	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.18.1d	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.18.2a	R	$2$	$2$	$0$	$0$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$
32.18.2b1	R	$2$	$2$	$0$	$0$	$-2$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$
32.18.2b2	R	$2$	$2$	$0$	$0$	$-2$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
32.18.2c1	R	$2$	$-2$	$0$	$0$	$0$	$-{\zeta }_{16}^{-2}-{\zeta }_{16}^2$	${\zeta }_{16}^{-2}+{\zeta }_{16}^2$	$-{\zeta }_{16}^{-3}-{\zeta }_{16}^3$	${\zeta }_{16}^{-1}+{\zeta }_{16}$	$-{\zeta }_{16}^{-1}-{\zeta }_{16}$	${\zeta }_{16}^{-3}+{\zeta }_{16}^3$
32.18.2c2	R	$2$	$-2$	$0$	$0$	$0$	$-{\zeta }_{16}^{-2}-{\zeta }_{16}^2$	${\zeta }_{16}^{-2}+{\zeta }_{16}^2$	${\zeta }_{16}^{-3}+{\zeta }_{16}^3$	$-{\zeta }_{16}^{-1}-{\zeta }_{16}$	${\zeta }_{16}^{-1}+{\zeta }_{16}$	$-{\zeta }_{16}^{-3}-{\zeta }_{16}^3$
32.18.2c3	R	$2$	$-2$	$0$	$0$	$0$	${\zeta }_{16}^{-2}+{\zeta }_{16}^2$	$-{\zeta }_{16}^{-2}-{\zeta }_{16}^2$	$-{\zeta }_{16}^{-1}-{\zeta }_{16}$	$-{\zeta }_{16}^{-3}-{\zeta }_{16}^3$	${\zeta }_{16}^{-3}+{\zeta }_{16}^3$	${\zeta }_{16}^{-1}+{\zeta }_{16}$
32.18.2c4	R	$2$	$-2$	$0$	$0$	$0$	${\zeta }_{16}^{-2}+{\zeta }_{16}^2$	$-{\zeta }_{16}^{-2}-{\zeta }_{16}^2$	${\zeta }_{16}^{-1}+{\zeta }_{16}$	${\zeta }_{16}^{-3}+{\zeta }_{16}^3$	$-{\zeta }_{16}^{-3}-{\zeta }_{16}^3$	$-{\zeta }_{16}^{-1}-{\zeta }_{16}$

Rational character table

		1A	2A	2B	2C	4A	8A	16A
	Size	1	1	8	8	2	4	8
	2 P	1A	1A	1A	1A	2A	4A	8A
32.18.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.18.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
32.18.1c		$1$	$1$	$-1$	$1$	$1$	$1$	$-1$
32.18.1d		$1$	$1$	$1$	$-1$	$1$	$1$	$-1$
32.18.2a		$2$	$2$	$0$	$0$	$2$	$-2$	$0$
32.18.2b		$4$	$4$	$0$	$0$	$-4$	$0$	$0$
32.18.2c		$8$	$-8$	$0$	$0$	$0$	$0$	$0$