Abstract group 256.55634: $C_4^2.C_2^4$

• Label: 256.55634
• Order: \( 2^{8} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{25} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{21} \cdot 3^{2} \)
• Perm deg.: $24$
• Trans deg.: $128$
• Rank: $6$

Group information

Description:	$C_4^2.C_2^4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^{14}.C_2^6.A_4.D_6.C_2$, of order \(301989888\)\(\medspace = 2^{25} \cdot 3^{2} \)
Composition factors:	$C_2$ x 8
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
Elements	1	31	224	256
Conjugacy classes	1	23	112	136
Divisions	1	23	80	104
Autjugacy classes	1	4	2	7

Dimension	1	2	4
Irr. complex chars.	128	0	8	136
Irr. rational chars.	64	32	8	104

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$128$
Rank:	$6$
Inequivalent generating 6-tuples:	$273420$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=e^{4}=f^{4}=[a,b]=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2,6,10)(3,11,13,7)(4,8,14,16)(5,12,15,9)(17,18,19,20)(21,22)(23,24), (1,3,6,13) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_4^2.C_2^2)$
Semidirect product:	$(Q_8\times C_2^3)$ $\,\rtimes\,$ $C_4$	$(C_4\times Q_8)$ $\,\rtimes\,$ $C_2^3$	$(C_4^2.C_2^3)$ $\,\rtimes\,$ $C_2$	$(C_4^2.C_2^2)$ $\,\rtimes\,$ $C_2^2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_2^3$	$C_2^4$ . $C_2^4$ (2)	$C_4^2$ . $C_2^4$	$C_2^3$ . $C_2^5$ (2)	all 38

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

Homology

Abelianization:	$C_{2}^{5} \times C_{4} $
Schur multiplier:	$C_{2}^{14}$
Commutator length:	$1$

Subgroups

There are 5903 subgroups in 5615 conjugacy classes, 5327 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^5\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2.C_2^4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4^2.C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^4$

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 256: $C_4^2.C_2^4$

Order 128: $C_4^2.C_2^3$ x 48, $C_4^2.C_2^3$ x 8, $C_4^2:C_2^3$ x 6, $Q_8\times C_2^4$

Order 64: $C_4^2.C_2^2$ x 256, $C_4^2.C_2^2$ x 192, $C_4^2:C_2^2$ x 144, $Q_8\times C_2^3$ x 60, $C_2^3.D_4$ x 12, $C_2^2\times C_4^2$ x 12, $C_2^4:C_4$ x 4, $C_2^4\times C_4$ x 3

Order 32: $C_2^2\times Q_8$ x 560, $C_4\times Q_8$ x 512, $C_4^2:C_2$ x 384, $C_2^2.D_4$ x 144, $C_2\times C_4^2$ x 144, $C_2^3\times C_4$ x 98, $C_2^3:C_4$ x 48, $C_2^5$

Order 16: $C_2\times Q_8$ x 960, $C_2^2\times C_4$ x 500, $C_4:C_4$ x 192, $C_4^2$ x 192, $C_2^2:C_4$ x 64, $C_2^4$ x 23

Order 8: $C_2\times C_4$ x 488, $Q_8$ x 256, $C_2^3$ x 99

Order 4: $C_2^2$ x 99, $C_4$ x 80

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_4^2.C_2^4$

Order 128: $Q_8\times C_2^4$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2:C_2^3$

Order 64: $Q_8\times C_2^3$ x 3, $C_2^4\times C_4$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_2^4:C_4$, $C_2^2\times C_4^2$

Order 32: $C_2^2\times Q_8$ x 4, $C_2^3\times C_4$ x 4, $C_2^5$, $C_4\times Q_8$, $C_4^2:C_2$, $C_2^2.D_4$, $C_2^3:C_4$, $C_2\times C_4^2$

Order 16: $C_2^2\times C_4$ x 6, $C_2^4$ x 4, $C_2\times Q_8$ x 3, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$

Order 8: $C_2^3$ x 7, $C_2\times C_4$ x 5, $Q_8$

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_4^2.C_2^4$ ($C_1$)

Order 128: $C_4^2.C_2^3$ ($C_2$) x 48, $C_4^2.C_2^3$ ($C_2$) x 8, $C_4^2:C_2^3$ ($C_2$) x 6, $Q_8\times C_2^4$ ($C_2$)

Order 64: $C_4^2.C_2^2$ ($C_2^2$) x 256, $C_4^2.C_2^2$ ($C_2^2$) x 192, $C_4^2:C_2^2$ ($C_2^2$) x 144, $Q_8\times C_2^3$ ($C_4$) x 32, $Q_8\times C_2^3$ ($C_2^2$) x 28, $C_2^3.D_4$ ($C_2^2$) x 12, $C_2^2\times C_4^2$ ($C_2^2$) x 12, $C_2^4:C_4$ ($C_2^2$) x 4, $C_2^4\times C_4$ ($C_2^2$) x 3

Order 32: $C_4\times Q_8$ ($C_2^3$) x 512, $C_2^2\times Q_8$ ($C_2\times C_4$) x 448, $C_4^2:C_2$ ($C_2^3$) x 384, $C_2^2.D_4$ ($C_2^3$) x 144, $C_2\times C_4^2$ ($C_2^3$) x 144, $C_2^2\times Q_8$ ($C_2^3$) x 112, $C_2^3\times C_4$ ($C_2^3$) x 50, $C_2^3\times C_4$ ($C_2\times C_4$) x 48, $C_2^3:C_4$ ($C_2^3$) x 48, $C_2^5$ ($C_2^3$)

Order 16: $C_2\times Q_8$ ($C_2^2\times C_4$) x 896, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 336, $C_4:C_4$ ($C_2^4$) x 192, $C_4^2$ ($C_2^4$) x 192, $C_2^2\times C_4$ ($C_2^4$) x 132, $C_2^2:C_4$ ($C_2^4$) x 64, $C_2\times Q_8$ ($C_2^4$) x 64, $C_2^4$ ($C_2^2\times C_4$) x 8, $C_2^4$ ($C_2^4$) x 7

Order 8: $C_2\times C_4$ ($C_2^3\times C_4$) x 336, $Q_8$ ($C_2^3\times C_4$) x 256, $C_2\times C_4$ ($C_2^5$) x 56, $C_2^3$ ($C_2^3\times C_4$) x 28, $C_2^3$ ($C_4.C_2^3$) x 8, $C_2^3$ ($C_2^5$) x 7

Order 4: $C_4$ ($C_2^4\times C_4$) x 48, $C_2^2$ ($C_4^2.C_2^2$) x 16, $C_2^2$ ($C_2^4\times C_4$) x 14, $C_2^2$ ($C_4.C_2^4$) x 12, $C_2^2$ ($C_2^6$)

Order 2: $C_2$ ($C_4^2.C_2^3$) x 12, $C_2$ ($C_4.C_2^5$) x 2, $C_2$ ($C_2^5\times C_4$)

Order 1: $C_1$ ($C_4^2.C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_4^2.C_2^4$ ($C_1$)

Order 128: $Q_8\times C_2^4$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_4^2:C_2^3$ ($C_2$)

Order 64: $Q_8\times C_2^3$ ($C_2^2$) x 2, $Q_8\times C_2^3$ ($C_4$), $C_2^4\times C_4$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2^4:C_4$ ($C_2^2$), $C_2^2\times C_4^2$ ($C_2^2$)

Order 32: $C_2^3\times C_4$ ($C_2^3$) x 3, $C_2^2\times Q_8$ ($C_2^3$) x 2, $C_2^2\times Q_8$ ($C_2\times C_4$) x 2, $C_2^5$ ($C_2^3$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_4\times Q_8$ ($C_2^3$), $C_4^2:C_2$ ($C_2^3$), $C_2^2.D_4$ ($C_2^3$), $C_2^3:C_4$ ($C_2^3$), $C_2\times C_4^2$ ($C_2^3$)

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

Order 8: $C_2^3$ ($C_2^5$) x 2, $C_2^3$ ($C_2^3\times C_4$) x 2, $C_2\times C_4$ ($C_2^5$) x 2, $C_2\times C_4$ ($C_2^3\times C_4$) x 2, $C_2^3$ ($C_4.C_2^3$), $Q_8$ ($C_2^3\times C_4$)

Order 4: $C_2^2$ ($C_2^4\times C_4$) x 2, $C_2^2$ ($C_2^6$), $C_2^2$ ($C_4.C_2^4$), $C_2^2$ ($C_4^2.C_2^2$), $C_4$ ($C_2^4\times C_4$)

Order 2: $C_2$ ($C_4.C_2^5$), $C_2$ ($C_2^5\times C_4$), $C_2$ ($C_4^2.C_2^3$)

Order 1: $C_1$ ($C_4^2.C_2^4$)

Series

Derived series

$C_4^2.C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_4^2.C_2^4$

$\rhd$

$C_4^2.C_2^3$

$\rhd$

$C_4^2.C_2^2$

$\rhd$

$C_4\times Q_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^2.C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^4$

$\lhd$

$C_4^2.C_2^4$

Character theory

Complex character table

See the $136 \times 136$ 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 $104 \times 104$ rational character table.