Abstract group 256.53354: $(C_2^2\times D_8):C_2^2$

• Label: 256.53354
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{15} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $36$
• Trans deg.: $64$
• Rank: $5$

Group information

Description:	$(C_2^2\times D_8):C_2^2$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^9.C_2^6$, of order \(32768\)\(\medspace = 2^{15} \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$3$
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	79	112	64	256
Conjugacy classes	1	23	28	12	64
Divisions	1	23	26	8	58
Autjugacy classes	1	9	10	3	23

Dimension	1	2	4	16
Irr. complex chars.	32	24	8	0	64
Irr. rational chars.	32	24	0	2	58

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$64$
Rank:	$5$
Inequivalent generating 5-tuples:	$9999360$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{8}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,2)(3,12)(4,16)(5,10)(6,20)(7,11)(8,14)(9,17)(13,27)(15,23)(18,31)(19,25) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8:D_4)$ $\,\rtimes\,$ $C_2^2$	$(D_4:D_4)$ $\,\rtimes\,$ $C_2^2$	$(D_4:D_4)$ $\,\rtimes\,$ $C_2^2$ (2)	$(D_4.D_4)$ $\,\rtimes\,$ $C_2^2$ (2)	all 45
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2\times Q_8)$ . $D_4$	$Q_8$ . $(C_2^2\times D_4)$ (2)	$(Q_8:C_4)$ . $C_2^3$ (3)	$(C_2\times Q_8)$ . $C_2^4$ (3)	all 45

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 4735 subgroups in 2391 conjugacy classes, 559 normal (49 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_2^2\times D_4$
Commutator:	$G' \simeq$ $C_2\times C_4$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2^2\times D_8):C_2^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2^2\times D_8):C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $D_4\times C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^2\times D_8):C_2^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 256: $(C_2^2\times D_8):C_2^2$

Order 128: $C_2^4.D_4$ x 8, $(C_2\times Q_8):D_4$ x 4, $(C_2\times D_4):D_4$ x 4, $(C_2^2\times Q_{16}):C_2$ x 2, $(C_2\times Q_8):D_4$ x 2, $(C_2\times Q_{16}):C_2^2$ x 2, $(C_2\times D_4):D_4$ x 2, $C_4.C_2^5$, $D_8:C_2^3$, $D_8:C_2^3$, $C_4^2.C_2^3$, $C_2^4.D_4$, $C_2^4.D_4$, $(C_2^3\times C_8):C_2$

Order 64: $D_8:C_2^2$ x 20, $C_4.C_2^4$ x 16, $D_4.D_4$ x 16, $D_4:D_4$ x 16, $C_2^3.D_4$ x 12, $D_4:C_2^3$ x 12, $(C_2^2\times C_8):C_2$ x 8, $D_8:C_2^2$ x 8, $C_2^2:Q_{16}$ x 8, $C_2^2:\SD_{16}$ x 8, $Q_8:D_4$ x 8, $D_4:D_4$ x 8, $D_4:C_2^3$ x 7, $C_4^2:C_2^2$ x 6, $C_4.C_2^4$ x 5, $C_2^3.D_4$ x 4, $C_2^3:C_8$ x 4, $Q_{16}:C_2^2$ x 4, $C_8:C_2^3$ x 4, $C_2^2.C_2^4$ x 4, $C_4^2.C_2^2$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3.D_4$ x 2, $C_2^2.Q_{16}$ x 2, $C_2^2.D_8$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_2^3:D_4$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_2^2\times Q_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$, $C_2^3\times C_8$, $C_4^2:C_2^2$

Order 32: $D_4:C_2^2$ x 194, $D_4:C_2^2$ x 80, $C_4.C_2^3$ x 48, $D_8:C_2$ x 48, $C_2^2\times D_4$ x 29, $C_2\times \SD_{16}$ x 24, $C_4:D_4$ x 20, $D_4:C_4$ x 16, $C_2^2:C_8$ x 16, $Q_{16}:C_2$ x 16, $D_8:C_2$ x 16, $Q_8:C_4$ x 16, $C_2^2\times Q_8$ x 12, $C_2\times Q_{16}$ x 12, $C_2\times D_8$ x 12, $C_2^2\times C_8$ x 12, $C_2^2:Q_8$ x 12, $C_2^3\times C_4$ x 10, $C_2\times \OD_{16}$ x 8, $C_2^2.D_4$ x 8, $C_2^2\wr C_2$ x 8, $C_4\times D_4$ x 8, $C_4^2:C_2$ x 6, $C_4^2:C_2$ x 4, $C_4:Q_8$ x 2, $C_4:D_4$ x 2, $C_2^2.D_4$ x 2, $C_2^3:C_4$, $C_2\times C_4^2$

Order 16: $D_4:C_2$ x 360, $C_2\times D_4$ x 245, $C_2^2\times C_4$ x 113, $C_2\times Q_8$ x 85, $\SD_{16}$ x 32, $C_2\times C_8$ x 24, $C_2^2:C_4$ x 20, $Q_{16}$ x 16, $D_8$ x 16, $C_4:C_4$ x 14, $C_2^4$ x 10, $\OD_{16}$ x 8, $C_4^2$ x 4

Order 8: $C_2\times C_4$ x 162, $D_4$ x 146, $C_2^3$ x 67, $Q_8$ x 50, $C_8$ x 8

Order 4: $C_2^2$ x 83, $C_4$ x 26

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $(C_2^2\times D_8):C_2^2$

Order 128: $C_4.C_2^5$, $D_8:C_2^3$, $D_8:C_2^3$, $C_4^2.C_2^3$, $(C_2\times Q_8):D_4$, $(C_2\times D_4):D_4$, $(C_2^2\times Q_{16}):C_2$, $(C_2\times Q_8):D_4$, $(C_2\times Q_{16}):C_2^2$, $(C_2\times D_4):D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $(C_2^3\times C_8):C_2$

Order 64: $D_4:C_2^3$ x 7, $C_2^3.D_4$ x 4, $D_4:C_2^3$ x 4, $D_8:C_2^2$ x 4, $(C_2^2\times C_8):C_2$ x 3, $C_4.C_2^4$ x 3, $C_4.C_2^4$ x 2, $Q_{16}:C_2^2$ x 2, $C_8:C_2^3$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $D_4.D_4$ x 2, $D_4:D_4$ x 2, $C_2^3.D_4$, $C_2^3.D_4$, $C_2^2.Q_{16}$, $C_2^2.D_8$, $C_2^3:C_8$, $D_8:C_2^2$, $C_2^2\times Q_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$, $C_2^3\times C_8$, $C_2^2.C_2^4$, $C_2^3:D_4$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^2:Q_{16}$, $C_2^2:\SD_{16}$, $Q_8:D_4$, $D_4:D_4$, $C_2^3.D_4$

Order 32: $D_4:C_2^2$ x 38, $C_2^2\times D_4$ x 14, $C_2^2\times Q_8$ x 7, $D_4:C_2^2$ x 6, $C_2^3\times C_4$ x 6, $C_2\times \SD_{16}$ x 6, $C_2^2\times C_8$ x 6, $C_4.C_2^3$ x 5, $C_4^2:C_2$ x 4, $D_4:C_4$ x 3, $D_8:C_2$ x 3, $C_2\times Q_{16}$ x 3, $C_2\times D_8$ x 3, $C_2\times \OD_{16}$ x 3, $C_4:D_4$ x 3, $Q_8:C_4$ x 3, $C_2^2:C_8$ x 2, $Q_{16}:C_2$ x 2, $D_8:C_2$ x 2, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_2^2:Q_8$ x 2, $C_2^2\wr C_2$ x 2, $C_4:Q_8$, $C_4:D_4$, $C_4\times D_4$, $C_2^2.D_4$, $C_2^3:C_4$, $C_2\times C_4^2$

Order 16: $C_2\times D_4$ x 38, $D_4:C_2$ x 36, $C_2^2\times C_4$ x 30, $C_2\times Q_8$ x 20, $C_2\times C_8$ x 8, $C_2^4$ x 6, $\SD_{16}$ x 4, $C_2^2:C_4$ x 4, $C_4:C_4$ x 3, $Q_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $C_4^2$ x 2

Order 8: $C_2\times C_4$ x 35, $D_4$ x 23, $C_2^3$ x 19, $Q_8$ x 12, $C_8$ x 3

Order 4: $C_2^2$ x 21, $C_4$ x 10

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $(C_2^2\times D_8):C_2^2$ ($C_1$)

Order 128: $C_2^4.D_4$ ($C_2$) x 8, $(C_2\times Q_8):D_4$ ($C_2$) x 4, $(C_2\times D_4):D_4$ ($C_2$) x 4, $(C_2^2\times Q_{16}):C_2$ ($C_2$) x 2, $(C_2\times Q_8):D_4$ ($C_2$) x 2, $(C_2\times Q_{16}):C_2^2$ ($C_2$) x 2, $(C_2\times D_4):D_4$ ($C_2$) x 2, $C_4.C_2^5$ ($C_2$), $D_8:C_2^3$ ($C_2$), $D_8:C_2^3$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $(C_2^3\times C_8):C_2$ ($C_2$)

Order 64: $D_4.D_4$ ($C_2^2$) x 16, $D_4:D_4$ ($C_2^2$) x 16, $C_2^3.D_4$ ($C_2^2$) x 12, $D_8:C_2^2$ ($C_2^2$) x 12, $(C_2^2\times C_8):C_2$ ($C_2^2$) x 8, $D_4:C_2^3$ ($C_2^2$) x 8, $C_2^2:Q_{16}$ ($C_2^2$) x 8, $C_2^2:\SD_{16}$ ($C_2^2$) x 8, $Q_8:D_4$ ($C_2^2$) x 8, $D_4:D_4$ ($C_2^2$) x 8, $C_2^3.D_4$ ($C_2^2$) x 4, $C_2^3:C_8$ ($C_2^2$) x 4, $C_4.C_2^4$ ($C_2^2$) x 4, $D_4:C_2^3$ ($C_2^2$) x 4, $Q_{16}:C_2^2$ ($C_2^2$) x 4, $C_8:C_2^3$ ($C_2^2$) x 4, $C_2^2.C_2^4$ ($C_2^2$) x 4, $C_4^2:C_2^2$ ($C_2^2$) x 4, $C_2^3.D_4$ ($C_2^2$) x 2, $C_2^2.Q_{16}$ ($C_2^2$) x 2, $C_2^2.D_8$ ($C_2^2$) x 2, $C_2^2\times \SD_{16}$ ($C_2^2$) x 2, $C_4^2.C_2^2$ ($C_2^2$) x 2, $C_4^2:C_2^2$ ($C_2^2$) x 2, $C_2^3.D_4$ ($C_2^2$) x 2, $C_2^2\times Q_{16}$ ($C_2^2$), $C_2^2\times D_8$ ($C_2^2$), $C_2^2\times \OD_{16}$ ($C_2^2$), $C_2^3\times C_8$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$)

Order 32: $D_4:C_2^2$ ($C_2^3$) x 24, $D_4:C_2^2$ ($D_4$) x 20, $D_4:C_4$ ($C_2^3$) x 16, $C_2^2:C_8$ ($C_2^3$) x 16, $C_2\times \SD_{16}$ ($C_2^3$) x 16, $Q_8:C_4$ ($C_2^3$) x 16, $C_2\times Q_{16}$ ($C_2^3$) x 8, $C_2\times D_8$ ($C_2^3$) x 8, $C_2^2\times C_8$ ($C_2^3$) x 8, $C_2^2:Q_8$ ($C_2^3$) x 8, $C_4:D_4$ ($C_2^3$) x 8, $C_2^2\times Q_8$ ($C_2^3$) x 7, $C_2^2\times D_4$ ($C_2^3$) x 7, $C_2\times \OD_{16}$ ($C_2^3$) x 4, $C_4^2:C_2$ ($C_2^3$) x 4, $C_2^2\times D_4$ ($D_4$) x 3, $C_2^3\times C_4$ ($C_2^3$) x 3, $C_2^2.D_4$ ($C_2^3$) x 2, $C_2^2\times Q_8$ ($D_4$)

Order 16: $D_4:C_2$ ($C_2\times D_4$) x 32, $C_2\times D_4$ ($C_2\times D_4$) x 20, $C_2^2\times C_4$ ($C_2\times D_4$) x 17, $C_2\times Q_8$ ($C_2\times D_4$) x 12, $C_2\times C_8$ ($C_2^4$) x 8, $C_2^2\times C_4$ ($C_2^4$) x 7, $C_2\times Q_8$ ($C_2^4$) x 6, $C_2\times D_4$ ($C_2^4$) x 6, $C_4:C_4$ ($C_2^4$) x 4, $C_2^4$ ($C_2\times D_4$) x 3

Order 8: $C_2\times C_4$ ($C_2^2\times D_4$) x 19, $C_2\times C_4$ ($C_2^2\wr C_2$) x 12, $Q_8$ ($C_2^2\times D_4$) x 8, $D_4$ ($C_2^2\times D_4$) x 8, $C_2^3$ ($C_2^2\times D_4$) x 7, $C_2^3$ ($C_2^2\wr C_2$) x 4, $C_2\times C_4$ ($C_2^5$)

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

Order 2: $C_2$ ($C_8.C_2^4$) x 2, $C_2$ ($C_2^4:D_4$)

Order 1: $C_1$ ($(C_2^2\times D_8):C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $(C_2^2\times D_8):C_2^2$ ($C_1$)

Order 128: $C_4.C_2^5$ ($C_2$), $D_8:C_2^3$ ($C_2$), $D_8:C_2^3$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $(C_2\times Q_8):D_4$ ($C_2$), $(C_2\times D_4):D_4$ ($C_2$), $(C_2^2\times Q_{16}):C_2$ ($C_2$), $(C_2\times Q_8):D_4$ ($C_2$), $(C_2\times Q_{16}):C_2^2$ ($C_2$), $(C_2\times D_4):D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $(C_2^3\times C_8):C_2$ ($C_2$)

Order 64: $D_4:C_2^3$ ($C_2^2$) x 5, $C_2^3.D_4$ ($C_2^2$) x 4, $(C_2^2\times C_8):C_2$ ($C_2^2$) x 3, $D_8:C_2^2$ ($C_2^2$) x 3, $C_4.C_2^4$ ($C_2^2$) x 2, $D_4:C_2^3$ ($C_2^2$) x 2, $Q_{16}:C_2^2$ ($C_2^2$) x 2, $C_8:C_2^3$ ($C_2^2$) x 2, $C_2^2\times \SD_{16}$ ($C_2^2$) x 2, $D_4.D_4$ ($C_2^2$) x 2, $D_4:D_4$ ($C_2^2$) x 2, $C_2^3.D_4$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2^2.Q_{16}$ ($C_2^2$), $C_2^2.D_8$ ($C_2^2$), $C_2^3:C_8$ ($C_2^2$), $C_2^2\times Q_{16}$ ($C_2^2$), $C_2^2\times D_8$ ($C_2^2$), $C_2^2\times \OD_{16}$ ($C_2^2$), $C_2^3\times C_8$ ($C_2^2$), $C_2^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_2^2:Q_{16}$ ($C_2^2$), $C_2^2:\SD_{16}$ ($C_2^2$), $Q_8:D_4$ ($C_2^2$), $D_4:D_4$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$)

Order 32: $D_4:C_2^2$ ($C_2^3$) x 10, $C_2^2\times C_8$ ($C_2^3$) x 5, $D_4:C_2^2$ ($D_4$) x 4, $C_2^2\times Q_8$ ($C_2^3$) x 4, $C_2^2\times D_4$ ($C_2^3$) x 4, $C_2\times \SD_{16}$ ($C_2^3$) x 4, $D_4:C_4$ ($C_2^3$) x 3, $Q_8:C_4$ ($C_2^3$) x 3, $C_2^2:C_8$ ($C_2^3$) x 2, $C_2^2\times D_4$ ($D_4$) x 2, $C_2^3\times C_4$ ($C_2^3$) x 2, $C_2\times Q_{16}$ ($C_2^3$) x 2, $C_2\times D_8$ ($C_2^3$) x 2, $C_2\times \OD_{16}$ ($C_2^3$) x 2, $C_4^2:C_2$ ($C_2^3$) x 2, $C_2^2\times Q_8$ ($D_4$), $C_2^2:Q_8$ ($C_2^3$), $C_4:D_4$ ($C_2^3$), $C_2^2.D_4$ ($C_2^3$)

Order 16: $C_2^2\times C_4$ ($C_2\times D_4$) x 7, $C_2\times D_4$ ($C_2\times D_4$) x 6, $D_4:C_2$ ($C_2\times D_4$) x 4, $C_2\times Q_8$ ($C_2\times D_4$) x 4, $C_2^2\times C_4$ ($C_2^4$) x 4, $C_2\times C_8$ ($C_2^4$) x 3, $C_2\times Q_8$ ($C_2^4$) x 3, $C_2\times D_4$ ($C_2^4$) x 3, $C_2^4$ ($C_2\times D_4$) x 2, $C_4:C_4$ ($C_2^4$)

Order 8: $C_2\times C_4$ ($C_2^2\times D_4$) x 10, $C_2^3$ ($C_2^2\times D_4$) x 4, $C_2\times C_4$ ($C_2^2\wr C_2$) x 3, $Q_8$ ($C_2^2\times D_4$) x 2, $D_4$ ($C_2^2\times D_4$) x 2, $C_2^3$ ($C_2^2\wr C_2$), $C_2\times C_4$ ($C_2^5$)

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

Order 2: $C_2$ ($C_8.C_2^4$), $C_2$ ($C_2^4:D_4$)

Order 1: $C_1$ ($(C_2^2\times D_8):C_2^2$)

Series

Derived series

$(C_2^2\times D_8):C_2^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$(C_2^2\times D_8):C_2^2$

$\rhd$

$(C_2^3\times C_8):C_2$

$\rhd$

$C_2^3:C_8$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times D_8):C_2^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_4:C_2^3$

$\lhd$

$(C_2^2\times D_8):C_2^2$

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $58 \times 58$ rational character table.