Abstract group 128.2136: $C_2^3\times C_{16}$

• Label: 128.2136
• Order: \( 2^{7} \)
• Exponent: \( 2^{4} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 7 \)
• Perm deg.: $22$
• Trans deg.: $128$
• Rank: $4$

Group information

Description:	$C_2^3\times C_{16}$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$C_4.C_2^7:\GL(3,2)$, of order \(86016\)\(\medspace = 2^{12} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	4	8	16
Elements	1	15	16	32	64	128
Conjugacy classes	1	15	16	32	64	128
Divisions	1	15	8	8	8	40
Autjugacy classes	1	2	2	2	1	8

Dimension	1	2	4	8
Irr. complex chars.	128	0	0	0	128
Irr. rational chars.	16	8	8	8	40

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$128$
Rank:	$4$
Inequivalent generating quadruples:	$960$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{16}=1 \rangle$
Permutation group:	Degree $22$ $\langle(7,22,14,18,10,20,12,16,8,21,13,17,9,19,11,15), (3,4), (5,6), (1,2), (7,14,10,12,8,13,9,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 15 & 18 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 4 \\ 8 & 17 \end{array}\right), \left(\begin{array}{rr} 17 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 16 \\ 16 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 17 & 16 \\ 16 & 17 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 13 & 12 \\ 8 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 24 \\ 16 & 25 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 22 \\ 10 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 2 & 5 \end{array}\right), \left(\begin{array}{rr} 9 & 22 \\ 10 & 15 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $C_{16}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $C_8$	$C_8$ . $C_2^4$	$(C_2^3\times C_8)$ . $C_2$	$(C_2^2\times C_8)$ . $C_4$	all 18
Aut. group:	$\Aut(C_{408})$

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

Homology

Primary decomposition:	$C_{2}^{3} \times C_{16}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$0$

Subgroups

There are 220 subgroups, all normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3\times C_{16}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2^3\times C_{16}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{16}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3\times C_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_{16}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 128: $C_2^3\times C_{16}$

Order 64: $C_2^2\times C_{16}$ x 14, $C_2^3\times C_8$

Order 32: $C_2\times C_{16}$ x 28, $C_2^2\times C_8$ x 14, $C_2^3\times C_4$

Order 16: $C_2\times C_8$ x 28, $C_2^2\times C_4$ x 14, $C_{16}$ x 8, $C_2^4$

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 15, $C_8$ x 8

Order 4: $C_2^2$ x 35, $C_4$ x 8

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $C_2^3\times C_{16}$

Order 64: $C_2^3\times C_8$, $C_2^2\times C_{16}$

Order 32: $C_2^2\times C_8$ x 2, $C_2^3\times C_4$, $C_2\times C_{16}$

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

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $C_2^3\times C_{16}$ ($C_1$)

Order 64: $C_2^2\times C_{16}$ ($C_2$) x 14, $C_2^3\times C_8$ ($C_2$)

Order 32: $C_2\times C_{16}$ ($C_2^2$) x 28, $C_2^2\times C_8$ ($C_2^2$) x 7, $C_2^2\times C_8$ ($C_4$) x 7, $C_2^3\times C_4$ ($C_4$)

Order 16: $C_2\times C_8$ ($C_2\times C_4$) x 21, $C_{16}$ ($C_2^3$) x 8, $C_2\times C_8$ ($C_2^3$) x 7, $C_2^2\times C_4$ ($C_2\times C_4$) x 7, $C_2^2\times C_4$ ($C_8$) x 7, $C_2^4$ ($C_8$)

Order 8: $C_2\times C_4$ ($C_2\times C_8$) x 21, $C_2^3$ ($C_{16}$) x 8, $C_2^3$ ($C_2\times C_8$) x 7, $C_2\times C_4$ ($C_2^2\times C_4$) x 7, $C_8$ ($C_2^2\times C_4$) x 7, $C_8$ ($C_2^4$)

Order 4: $C_2^2$ ($C_2\times C_{16}$) x 28, $C_2^2$ ($C_2^2\times C_8$) x 7, $C_4$ ($C_2^2\times C_8$) x 7, $C_4$ ($C_2^3\times C_4$)

Order 2: $C_2$ ($C_2^2\times C_{16}$) x 14, $C_2$ ($C_2^3\times C_8$)

Order 1: $C_1$ ($C_2^3\times C_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $C_2^3\times C_{16}$ ($C_1$)

Order 64: $C_2^3\times C_8$ ($C_2$), $C_2^2\times C_{16}$ ($C_2$)

Order 32: $C_2^3\times C_4$ ($C_4$), $C_2^2\times C_8$ ($C_2^2$), $C_2^2\times C_8$ ($C_4$), $C_2\times C_{16}$ ($C_2^2$)

Order 16: $C_2\times C_8$ ($C_2^3$), $C_2\times C_8$ ($C_2\times C_4$), $C_2^4$ ($C_8$), $C_2^2\times C_4$ ($C_2\times C_4$), $C_2^2\times C_4$ ($C_8$), $C_{16}$ ($C_2^3$)

Order 8: $C_2^3$ ($C_2\times C_8$), $C_2^3$ ($C_{16}$), $C_2\times C_4$ ($C_2\times C_8$), $C_2\times C_4$ ($C_2^2\times C_4$), $C_8$ ($C_2^4$), $C_8$ ($C_2^2\times C_4$)

Order 4: $C_2^2$ ($C_2^2\times C_8$), $C_2^2$ ($C_2\times C_{16}$), $C_4$ ($C_2^3\times C_4$), $C_4$ ($C_2^2\times C_8$)

Order 2: $C_2$ ($C_2^3\times C_8$), $C_2$ ($C_2^2\times C_{16}$)

Order 1: $C_1$ ($C_2^3\times C_{16}$)

Series

Derived series

$C_2^3\times C_{16}$

$\rhd$

$C_1$

Chief series

$C_2^3\times C_{16}$

$\rhd$

$C_2^2\times C_{16}$

$\rhd$

$C_2\times C_{16}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3\times C_{16}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_{16}$

Supergroups

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

This group is a maximal quotient of 18 larger groups in the database.

Character theory

Complex character table

See the $128 \times 128$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.