Abstract group 64.186: $C_2\times D_{16}$

• Label: 64.186
• Order: \( 2^{6} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $18$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$C_2\times D_{16}$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$(C_2^2\times C_8).C_2^5$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 6
Nilpotency class:	$4$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16
Elements	1	35	4	8	16	64
Conjugacy classes	1	7	2	4	8	22
Divisions	1	7	2	2	2	14
Autjugacy classes	1	3	2	2	1	9

Dimension	1	2	4	8
Irr. complex chars.	8	14	0	0	22
Irr. rational chars.	8	2	2	2	14

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$32$
Rank:	$3$
Inequivalent generating triples:	$84$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	3	9

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{16}=[a,b]=[a,c]=1, c^{b}=c^{15} \rangle$
Permutation group:	Degree $18$ $\langle(1,2)(3,7)(4,6)(5,13)(8,12)(9,11)(10,14)(15,16)(17,18), (2,6)(4,12)(5,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 15 & 12 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 17 & 18 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 31 & 14 \\ 14 & 3 \end{array}\right), \left(\begin{array}{rr} 11 & 4 \\ 4 & 23 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 33 \end{array}\right), \left(\begin{array}{rr} 1 & 17 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/34\Z)$
Transitive group:	32T297				more information
Direct product:	$C_2$ $\, \times\, $ $D_{16}$
Semidirect product:	$D_8$ $\,\rtimes\,$ $C_2^2$ (4)	$C_{16}$ $\,\rtimes\,$ $C_2^2$ (2)	$(C_2\times D_8)$ $\,\rtimes\,$ $C_2$ (2)	$(C_2\times C_{16})$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $D_4$	$C_4$ . $D_8$	$C_2^2$ . $D_8$	$C_8$ . $C_2^3$	all 8

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

Homology

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

Subgroups

There are 137 subgroups in 49 conjugacy classes, 25 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_8$
Commutator:	$G' \simeq$ $C_8$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times D_{16}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times D_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_{16}$

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 64: $C_2\times D_{16}$

Order 32: $D_{16}$ x 4, $C_2\times D_8$ x 2, $C_2\times C_{16}$

Order 16: $D_8$ x 6, $C_2\times D_4$ x 2, $C_{16}$ x 2, $C_2\times C_8$

Order 8: $D_4$ x 6, $C_2^3$ x 2, $C_8$ 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 64: $C_2\times D_{16}$

Order 32: $C_2\times D_8$, $D_{16}$, $C_2\times C_{16}$

Order 16: $D_8$ x 2, $C_2\times C_8$, $C_2\times D_4$, $C_{16}$

Order 8: $D_4$ x 2, $C_8$ x 2, $C_2^3$, $C_2\times C_4$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $C_2\times D_{16}$ ($C_1$)

Order 32: $D_{16}$ ($C_2$) x 4, $C_2\times D_8$ ($C_2$) x 2, $C_2\times C_{16}$ ($C_2$)

Order 16: $D_8$ ($C_2^2$) x 4, $C_{16}$ ($C_2^2$) x 2, $C_2\times C_8$ ($C_2^2$)

Order 8: $C_2\times C_4$ ($D_4$), $C_8$ ($C_2^3$), $C_8$ ($D_4$)

Order 4: $C_2^2$ ($D_8$), $C_4$ ($D_8$), $C_4$ ($C_2\times D_4$)

Order 2: $C_2$ ($D_{16}$) x 2, $C_2$ ($C_2\times D_8$)

Order 1: $C_1$ ($C_2\times D_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $C_2\times D_{16}$ ($C_1$)

Order 32: $C_2\times D_8$ ($C_2$), $D_{16}$ ($C_2$), $C_2\times C_{16}$ ($C_2$)

Order 16: $D_8$ ($C_2^2$), $C_2\times C_8$ ($C_2^2$), $C_{16}$ ($C_2^2$)

Order 8: $C_2\times C_4$ ($D_4$), $C_8$ ($C_2^3$), $C_8$ ($D_4$)

Order 4: $C_2^2$ ($D_8$), $C_4$ ($D_8$), $C_4$ ($C_2\times D_4$)

Order 2: $C_2$ ($C_2\times D_8$), $C_2$ ($D_{16}$)

Order 1: $C_1$ ($C_2\times D_{16}$)

Series

Derived series

$C_2\times D_{16}$

$\rhd$

$C_8$

$\rhd$

$C_1$

Chief series

$C_2\times D_{16}$

$\rhd$

$D_{16}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times D_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2\times D_{16}$

Supergroups

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

This group is a maximal quotient of 62 larger groups in the database.

Character theory

Complex character table

See the $22 \times 22$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	2F	2G	4A	4B	8A	8B	16A	16B
	Size	1	1	1	1	8	8	8	8	2	2	4	4	8	8
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	2C	2C	4B	4B	8B	8B
64.186.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.186.1b		$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
64.186.1c		$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$
64.186.1d		$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$
64.186.1e		$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
64.186.1f		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
64.186.1g		$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
64.186.1h		$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
64.186.2a		$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$-2$	$2$	$2$	$-2$	$0$	$0$
64.186.2b		$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$2$	$2$	$-2$	$-2$	$0$	$0$
64.186.2c		$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$	$4$	$-4$	$0$	$0$	$0$	$0$
64.186.2d		$4$	$4$	$4$	$4$	$0$	$0$	$0$	$0$	$-4$	$-4$	$0$	$0$	$0$	$0$
64.186.2e		$8$	$-8$	$-8$	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.186.2f		$8$	$-8$	$8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$