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C₄²×D₄ #2271

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₄²×D₄

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²
commutator	C₂	C₂²×C₄²
Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 12 points:

(1,2,3,4);(5,6,7,8);(9,10,11,12);(10,12)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₄×C₄²⋊C₃, #11711184, #11711187, …
maximal subgroup of	C₄×D₄⋊C₈, #8170, #8172, …
maximal normal subgroup of	C₄×D₄⋊C₈, #8170, #8172, …
maximal quotient of	C₄²×C₂²⋊C₄, C₄²×C₄⋊C₄, C₄×#386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1003);