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C₂×D₄⋊C₈ #1474

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×#72214271 (order 2048) · Inn(G) = D₄ · Out(G) = C₂×#2403
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂×D₄⋊C₈

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₄
commutator	C₄	C₂²×C₈
Frattini, cosocle	C₄²	C₂³
socle	C₂³	C₂²⋊C₄
prime radical	C₂²×D₄	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,13,15,14,16,17,18);(4,7)(5,10)(8,9)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×D₄⋊C₈, #8170, #8172, …
maximal normal subgroup of	C₄×D₄⋊C₈, #8170, #8172, …
maximal quotient of	C₂×#1270, C₂×#1271, #7560, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 206);