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C₂×Q₈×D₇ #6549

order	224
automorphism	Aut(G) order 32256
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×Q₈×D₇

Splittings

Fitting	C₂×C₇×Q₈ ⋊ C₂
Sylow	C₇ ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂⁴
Frattini	C₂	C₂³×D₇
Fitting	C₂×C₇×Q₈	C₂
socle	C₂²×C₇	C₂³
prime radical	C₂²×D₇	C₂²
hypercenter	C₂×Q₈	D₇
hypocenter	C₇	C₂²×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#101, D₇×#103, #89574, …
maximal normal subgroup of	D₇×#101, D₇×#103, #89574, …
maximal quotient of	C₂×#6450, C₂×#6451, C₂×#6452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 181);