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D₄×Dic₇ #6497

order	224
automorphism	Aut(G) = C₂²×#120×AGL(1,𝔽₇) (order 5376) · Inn(G) = C₂²×D₇ · Out(G) = C₂²×C₃×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

D₄×Dic₇

Splittings

C₇ ⋊ C₄×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄
Fitting, prime radical	C₂×C₇×D₄	C₂
socle, cosocle	C₂²×C₇	C₂³
hypercenter	C₂×D₄	D₇
hypocenter	C₇	C₄×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#89528, #89529, #89530, …
maximal normal subgroup of	#89528, #89529, #89530, …
maximal quotient of	C₂²⋊C₄×Dic₇, #89722, #89723, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 129);