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C₇×#104 #6421

order	224
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂³×C₃×D₄ (order 192) · Inn(G) = D₄ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

Subgroups

	subgroup	G/H
center	C₄×C₇	D₄
commutator	C₄	C₂×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂²×C₇
socle	C₂×C₇	C₂²⋊C₄
prime radical	C₇×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ G

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#104, #89593, #89594, …
maximal subgroup of	D₇×#104, #89593, #89594, …
maximal normal subgroup of	D₇×#104, #89593, #89594, …
maximal quotient of	C₇×D₄⋊C₈, C₇×Q₈⋊C₈, C₇×#327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 53);