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C₂³×Dic₇ #6555

order	224
isoclinism	isoclinic class of D₇
automorphism	Aut(G) order 903168
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×Dic₇

Splittings

C₇ ⋊ C₂³×C₄

Subgroups

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

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×Dic₇, #89714, #89715, …
maximal normal subgroup of	C₂²⋊C₄×Dic₇, #89714, #89715, …
maximal quotient of	C₂³×C₇⋊C₈, C₂²×#1136, C₂²×C₄×Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 187);