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C₂²×C₈×D₇ #90427

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×C₈×D₇

Splittings

Fitting	C₂²×C₇×C₈ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂³×C₈

Subgroups

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

338 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)(3,4)(6,7)(8,9)(10,11);(5,6,8,10,11,9,7)(12,13,14,16,15,17,18,19);(3,4);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726401, #11726402, D₇×#336, …
maximal normal subgroup of	#11726401, #11726402, D₇×#336, …
maximal quotient of	C₂×C₈×Dic₁₄, C₈×#1157, C₂×C₄×C₈×D₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1189);