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C₂×C₈×Dic₇ #89870

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₂×C₈×Dic₇

Splittings

C₇ ⋊ C₂×C₄×C₈

Subgroups

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

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2)(3,4,5,6)(8,9)(10,11)(12,13);(7,8,10,12,13,11,9)(14,15,16,18,17,19,20,21);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725610, #11725616, #11726350, …
maximal normal subgroup of	#11725610, #11725616, #11726350, …
maximal quotient of	C₂×C₈×C₇⋊C₈, C₂×#89249, C₈×#1136, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 632);