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C₂²×#1132 #90430

order	448
isoclinism	isoclinic class of #1132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

Fitting	C₂²×C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ C₂²×SD₁₆

Subgroups

	subgroup	G/H
center	C₂³	D₂₈
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₂²×C₇×C₈	C₂
socle	C₂³×C₇	D₄
prime radical	C₂²×D₂₈	C₂
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	C₂²×SD₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

298 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:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726426, #11726429, #11726430, …
maximal normal subgroup of	#11726426, #11726429, #11726430, …
maximal quotient of	C₂×#89452, C₂×#89466, C₂×#89465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1192);