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Q₈×D₂₈ #90266

order	448
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Q₈×D₂₈

Splittings

Fitting	C₄×C₇×Q₈ ⋊ C₂
Sylow	C₇ ⋊ D₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₄×C₇×Q₈	C₂
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	D₄×Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,4,6,7,5,3)(8,9,10,11);(2,3)(4,5)(6,7)(9,11);(12,14,15,19)(13,16,17,18);(12,13,15,17)(14,18,19,16)

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726005, #11726020, #11726023, …
maximal normal subgroup of	#11726005, #11726020, #11726023, …
maximal quotient of	#11731095, #11731120, #11731127, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1028);