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D₇×C₂²⋊C₄ #6443

order	224
automorphism	Aut(G) = AGL(1,𝔽₇)×#2846 (order 5376) · Inn(G) = C₂²×D₇ · Out(G) = C₃×#120
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

D₇×C₂²⋊C₄

Splittings

Fitting	C₇×C₂²⋊C₄ ⋊ C₂
Sylow	C₇ ⋊ C₂×C₂²⋊C₄

Subgroups

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

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#99, C₂×D₇×C₂²⋊C₄, #90177, …
maximal normal subgroup of	D₇×#99, C₂×D₇×C₂²⋊C₄, #90177, …
maximal quotient of	#89418, D₇×#95, #89436, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 75);