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C₂×C₄⋊Dic₇ #6488

order	224
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) order 21504
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₄⋊Dic₇

Splittings

C₇ ⋊ C₂×C₄⋊C₄

Subgroups

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

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#89324, #89346, #89349, …
maximal normal subgroup of	#89324, #89346, #89349, …
maximal quotient of	C₂×C₄⋊(C₇⋊C₈), #89696, C₄×C₄⋊Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 120);