Abstract group 64.27: $C_{16}:C_4$

• Label: 64.27
• Order: \( 2^{6} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $20$
• Trans deg.: $64$
• Rank: $2$

Group information

Description:	$C_{16}:C_4$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$C_2^5.D_4$, of order \(256\)\(\medspace = 2^{8} \)
Composition factors:	$C_2$ x 6
Nilpotency class:	$2$
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	3	12	16	32	64
Conjugacy classes	1	3	8	12	16	40
Divisions	1	3	4	4	4	16
Autjugacy classes	1	2	3	2	1	9

Dimension	1	2	4	8
Irr. complex chars.	32	8	0	0	40
Irr. rational chars.	4	6	4	2	16

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$64$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	10

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{16}=1, b^{a}=b^{9} \rangle$
Permutation group:	Degree $20$ $\langle(1,2,4,8,6,7,13,14,5,9,12,15,3,10,11,16)(17,18,19,20), (1,3,5,6)(2,7,9,10) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{lll}1 & \alpha^{3} & \alpha^{3} \\ \alpha^{5} & \alpha^{4} & \alpha \\ \alpha & \alpha^{5} & \alpha^{7} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{7} & \alpha^{3} & \alpha^{7} \\ \alpha^{3} & \alpha^{3} & \alpha^{7} \\ \alpha^{6} & \alpha^{4} & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{6} & \alpha^{3} & \alpha^{3} \\ \alpha^{5} & \alpha^{5} & \alpha \\ \alpha & \alpha^{5} & 0 \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{4} & 0 & 0 \\ 0 & \alpha^{4} & 0 \\ 0 & 0 & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{lll}1 & 1 & 1 \\ \alpha^{2} & \alpha^{3} & \alpha^{6} \\ \alpha^{6} & \alpha^{2} & \alpha \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{2} & \alpha^{4} & \alpha^{3} \\ 1 & 1 & 0 \\ 1 & \alpha & 0 \\ \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{9}) = \GL_{3}(\F_{3}[\alpha]/(\alpha^{2} + 2 \alpha + 2))$
	$\left\langle \left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 13 & 4 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{16}$ $\,\rtimes\,$ $C_4$ (4)				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $C_8$ (2)	$C_4^2$ . $C_4$	$C_4$ . $C_4^2$	$C_2$ . $\OD_{32}$ (2)	all 14

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

Homology

Abelianization:	$C_{4} \times C_{8} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 29 subgroups in 27 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:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_4\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{16}:C_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{16}:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}:C_4$

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 64: $C_{16}:C_4$

Order 32: $C_2\times C_{16}$ x 2, $C_4\times C_8$

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 2, $C_4^2$

Order 8: $C_8$ x 4, $C_2\times C_4$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $C_{16}:C_4$

Order 32: $C_4\times C_8$, $C_2\times C_{16}$

Order 16: $C_2\times C_8$ x 2, $C_4^2$, $C_{16}$

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $C_{16}:C_4$ ($C_1$)

Order 32: $C_2\times C_{16}$ ($C_2$) x 2, $C_4\times C_8$ ($C_2$)

Order 16: $C_{16}$ ($C_4$) x 4, $C_2\times C_8$ ($C_2^2$), $C_2\times C_8$ ($C_4$), $C_4^2$ ($C_4$)

Order 8: $C_2\times C_4$ ($C_8$) x 2, $C_8$ ($C_2\times C_4$) x 2, $C_8$ ($C_8$) x 2, $C_2\times C_4$ ($C_2\times C_4$)

Order 4: $C_2^2$ ($C_2\times C_8$), $C_4$ ($C_2\times C_8$), $C_4$ ($C_4^2$)

Order 2: $C_2$ ($\OD_{32}$) x 2, $C_2$ ($C_4\times C_8$)

Order 1: $C_1$ ($C_{16}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $C_{16}:C_4$ ($C_1$)

Order 32: $C_4\times C_8$ ($C_2$), $C_2\times C_{16}$ ($C_2$)

Order 16: $C_2\times C_8$ ($C_2^2$), $C_2\times C_8$ ($C_4$), $C_4^2$ ($C_4$), $C_{16}$ ($C_4$)

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

Order 4: $C_2^2$ ($C_2\times C_8$), $C_4$ ($C_2\times C_8$), $C_4$ ($C_4^2$)

Order 2: $C_2$ ($C_4\times C_8$), $C_2$ ($\OD_{32}$)

Order 1: $C_1$ ($C_{16}:C_4$)

Series

Derived series

$C_{16}:C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{16}:C_4$

$\rhd$

$C_2\times C_{16}$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_{16}:C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_{16}:C_4$

Supergroups

This group is a maximal subgroup of 77 larger groups in the database.

This group is a maximal quotient of 56 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	4C	4D	8A	8B	8C	8D	16A	16B	16C	16D
	Size	1	1	1	1	2	2	4	4	4	4	4	4	8	8	8	8
	2 P	1A	1A	1A	1A	2C	2C	2A	2B	4B	4B	4A	4A	8A	8A	8B	8B
64.27.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.27.1b		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
64.27.1c		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
64.27.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
64.27.1e		$2$	$-2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$2$	$0$
64.27.1f		$2$	$-2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$-2$	$0$
64.27.1g		$2$	$-2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$2$	$2$	$0$	$0$	$-2$	$2$	$0$	$2$
64.27.1h		$2$	$-2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$2$	$2$	$0$	$0$	$2$	$-2$	$0$	$-2$
64.27.1i		$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$-2$	$-2$	$2$	$2$	$0$	$0$	$0$	$0$
64.27.1j		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$
64.27.1k		$4$	$-4$	$-4$	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$-4$	$4$	$0$	$0$	$0$	$0$
64.27.1l		$4$	$-4$	$-4$	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$4$	$-4$	$0$	$0$	$0$	$0$
64.27.1m		$4$	$4$	$4$	$4$	$-4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.27.1n		$4$	$4$	$4$	$4$	$-4$	$-4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.27.2a		$8$	$-8$	$8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.27.2b		$8$	$8$	$-8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$