Abstract group 128.1655: $C_4^2.C_2^3$

• Label: 128.1655
• Order: \( 2^{7} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $24$
• Trans deg.: $64$
• Rank: $4$

Group information

Description:	$C_4^2.C_2^3$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^6.D_4^2$, of order \(4096\)\(\medspace = 2^{12} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$2$
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
Elements	1	15	48	64	128
Conjugacy classes	1	7	20	16	44
Divisions	1	7	14	8	30
Autjugacy classes	1	4	6	2	13

Dimension	1	2	4	8
Irr. complex chars.	32	8	4	0	44
Irr. rational chars.	16	8	4	2	30

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$64$
Rank:	$4$
Inequivalent generating quadruples:	$20160$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{8}=[a,c]=[a,d]=[b,c]=1, b^{a}=bd^{4}, d^{b}=c^{2}d, d^{c}=d^{5} \rangle$
Permutation group:	Degree $24$ $\langle(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18,20,22)(19,21,23,24), (3,9) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4.C_4^2)$ $\,\rtimes\,$ $C_2$	$(C_4^2.C_4)$ $\,\rtimes\,$ $C_2$	$(C_4\times \OD_{16})$ $\,\rtimes\,$ $C_2$	$(C_4^2.C_2^2)$ $\,\rtimes\,$ $C_2$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $C_2^3$ (3)	$(C_4\times Q_8)$ . $C_4$	$(C_4\times D_4)$ . $C_4$ (2)	$C_4$ . $(D_4:C_2^2)$	all 32

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{3} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 268 subgroups in 192 conjugacy classes, 132 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2.C_2^3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4^2.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^3\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^3$

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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

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

Classes of subgroups up to conjugation

Order 128: $C_4^2.C_2^3$

Order 64: $C_4^2.C_2^2$ x 4, $C_4^2.C_4$ x 4, $(C_2^2\times C_8):C_2$ x 2, $C_4^2.C_2^2$ x 2, $C_4\times \OD_{16}$, $C_4.C_4^2$, $C_4^2.C_2^2$

Order 32: $C_2^2:C_8$ x 8, $C_4:C_8$ x 8, $C_8:C_4$ x 6, $C_4\times D_4$ x 6, $C_4^2:C_2$ x 3, $C_2\times C_4^2$ x 3, $C_2\times \OD_{16}$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$ x 2, $C_4\times Q_8$ x 2, $D_4:C_2^2$

Order 16: $C_2\times C_8$ x 12, $C_4^2$ x 10, $C_2^2\times C_4$ x 9, $C_4:C_4$ x 6, $C_2^2:C_4$ x 6, $\OD_{16}$ x 4, $D_4:C_2$ x 4, $C_2\times D_4$ x 3, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 26, $C_8$ x 8, $D_4$ x 6, $C_2^3$ x 3, $Q_8$ x 2

Order 4: $C_4$ x 14, $C_2^2$ x 11

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $C_4^2.C_2^3$

Order 64: $(C_2^2\times C_8):C_2$, $C_4\times \OD_{16}$, $C_4.C_4^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2.C_4$, $C_4^2.C_2^2$

Order 32: $C_8:C_4$ x 3, $C_4\times D_4$ x 2, $C_4^2:C_2$ x 2, $C_2\times C_4^2$ x 2, $C_2^2:C_8$, $D_4:C_2^2$, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4\times C_8$, $C_4\times Q_8$, $C_4:C_8$

Order 16: $C_4^2$ x 5, $C_2^2\times C_4$ x 4, $C_2\times C_8$ x 3, $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_2\times D_4$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 10, $C_2^3$ x 2, $D_4$ x 2, $C_8$ x 2, $Q_8$

Order 4: $C_4$ x 6, $C_2^2$ x 5

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $C_4^2.C_2^3$ ($C_1$)

Order 64: $C_4^2.C_2^2$ ($C_2$) x 4, $C_4^2.C_4$ ($C_2$) x 4, $(C_2^2\times C_8):C_2$ ($C_2$) x 2, $C_4^2.C_2^2$ ($C_2$) x 2, $C_4\times \OD_{16}$ ($C_2$), $C_4.C_4^2$ ($C_2$), $C_4^2.C_2^2$ ($C_2$)

Order 32: $C_2^2:C_8$ ($C_2^2$) x 8, $C_4:C_8$ ($C_2^2$) x 8, $C_8:C_4$ ($C_2^2$) x 6, $C_4\times D_4$ ($C_4$) x 6, $C_4^2:C_2$ ($C_2^2$) x 3, $C_2\times C_4^2$ ($C_2^2$) x 3, $C_2\times \OD_{16}$ ($C_2^2$) x 2, $C_2^2\times C_8$ ($C_2^2$) x 2, $C_4\times C_8$ ($C_2^2$) x 2, $C_4\times Q_8$ ($C_4$) x 2, $D_4:C_2^2$ ($C_2^2$)

Order 16: $C_2\times C_8$ ($C_2^3$) x 8, $C_4:C_4$ ($C_2\times C_4$) x 6, $C_2^2:C_4$ ($C_2\times C_4$) x 6, $C_4^2$ ($C_2\times C_4$) x 6, $C_2^2\times C_4$ ($C_2\times C_4$) x 6, $C_4^2$ ($C_2^3$) x 4, $C_2\times D_4$ ($C_2\times C_4$) x 3, $C_2^2\times C_4$ ($C_2^3$) x 3, $C_2\times Q_8$ ($C_2\times C_4$)

Order 8: $C_2\times C_4$ ($C_2^2\times C_4$) x 11, $C_2\times C_4$ ($D_4:C_2$) x 4, $C_2^3$ ($C_2^2\times C_4$) x 3, $C_2\times C_4$ ($C_2^4$)

Order 4: $C_2^2$ ($C_4^2:C_2$) x 2, $C_4$ ($D_4:C_2^2$) x 2, $C_4$ ($C_4^2:C_2$) x 2, $C_2^2$ ($C_2^3\times C_4$)

Order 2: $C_2$ ($C_8.C_2^3$) x 2, $C_2$ ($C_4^2:C_2^2$)

Order 1: $C_1$ ($C_4^2.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $C_4^2.C_2^3$ ($C_1$)

Order 64: $(C_2^2\times C_8):C_2$ ($C_2$), $C_4\times \OD_{16}$ ($C_2$), $C_4.C_4^2$ ($C_2$), $C_4^2.C_2^2$ ($C_2$), $C_4^2.C_2^2$ ($C_2$), $C_4^2.C_4$ ($C_2$), $C_4^2.C_2^2$ ($C_2$)

Order 32: $C_8:C_4$ ($C_2^2$) x 3, $C_4\times D_4$ ($C_4$) x 2, $C_4^2:C_2$ ($C_2^2$) x 2, $C_2\times C_4^2$ ($C_2^2$) x 2, $C_2^2:C_8$ ($C_2^2$), $D_4:C_2^2$ ($C_2^2$), $C_2\times \OD_{16}$ ($C_2^2$), $C_2^2\times C_8$ ($C_2^2$), $C_4\times C_8$ ($C_2^2$), $C_4\times Q_8$ ($C_4$), $C_4:C_8$ ($C_2^2$)

Order 16: $C_4^2$ ($C_2^3$) x 3, $C_2\times C_8$ ($C_2^3$) x 2, $C_4:C_4$ ($C_2\times C_4$) x 2, $C_2^2:C_4$ ($C_2\times C_4$) x 2, $C_4^2$ ($C_2\times C_4$) x 2, $C_2\times D_4$ ($C_2\times C_4$) x 2, $C_2^2\times C_4$ ($C_2^3$) x 2, $C_2^2\times C_4$ ($C_2\times C_4$) x 2, $C_2\times Q_8$ ($C_2\times C_4$)

Order 8: $C_2\times C_4$ ($C_2^2\times C_4$) x 5, $C_2^3$ ($C_2^2\times C_4$) x 2, $C_2\times C_4$ ($C_2^4$), $C_2\times C_4$ ($D_4:C_2$)

Order 4: $C_2^2$ ($C_2^3\times C_4$), $C_2^2$ ($C_4^2:C_2$), $C_4$ ($D_4:C_2^2$), $C_4$ ($C_4^2:C_2$)

Order 2: $C_2$ ($C_8.C_2^3$), $C_2$ ($C_4^2:C_2^2$)

Order 1: $C_1$ ($C_4^2.C_2^3$)

Series

Derived series

$C_4^2.C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_4^2.C_2^3$

$\rhd$

$C_4^2.C_4$

$\rhd$

$C_8:C_4$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^2.C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2.C_2^3$

Supergroups

This group is a maximal subgroup of 82 larger groups in the database.

This group is a maximal quotient of 79 larger groups in the database.

Character theory

Complex character table

See the $44 \times 44$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.