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C₂²×Q₈×D₇ #90610

order	448
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×Q₈×D₇

Splittings

Fitting	C₂²×C₇×Q₈ ⋊ C₂
Sylow	C₇ ⋊ C₂³×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂⁵
Frattini	C₂	C₂⁴×D₇
Fitting	C₂²×C₇×Q₈	C₂
socle	C₂³×C₇	C₂³
prime radical	C₂³×D₇	C₂²
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	C₂³×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 19 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11727867, #11728020, #11733944, …
maximal normal subgroup of	#11727867, #11728020, #11733944, …
maximal quotient of	C₂²×#6450, C₂²×#6452, C₂²×#6451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1372);