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C₄×C₄⋊Dic₇ #89706

order	448
isoclinism	isoclinic class of Dic₁₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₄×C₄⋊Dic₇

Splittings

C₇ ⋊ C₄×C₄⋊C₄

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₇
commutator	C₂×C₇	C₂×C₄²
Frattini	C₂³	C₂²×D₇
Fitting	C₂×C₄²×C₇	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2,4,6,7,5,3)(16,17,18,19);(2,3)(4,5)(6,7)(12,13,14,15)(17,19);(8,9,10,11)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725461, #11725510, #11725521, …
maximal normal subgroup of	#11725461, #11725510, #11725521, …
maximal quotient of	C₄×C₄⋊(C₇⋊C₈), #11727145, #11727161, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 468);