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C₂×C₃²×Dic₇ #92705

order	504
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₂×C₃²×Dic₇

Splittings

derived, Sylow	C₇ ⋊ C₂×C₃²×C₄
Hall	C₃²×C₇ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₃²	D₇
commutator, hypocenter	C₇	C₂×C₃²×C₄
Frattini	C₂	C₂×C₃²×D₇
Fitting, socle, prime radical	C₂²×C₃²×C₇	C₂
cosocle	C₂×C₇	C₂²×C₃²
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₃²

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×Dic₃×Dic₇, C₃×(C₂×D₃)⋊Dic₇, C₃×#66575, …
maximal normal subgroup of	C₃×Dic₃×Dic₇, C₃×(C₂×D₃)⋊Dic₇, C₃×#66575, …
maximal quotient of	C₂×C₃²×C₇⋊C₈, C₃²×#1136, C₃²×C₄×Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 131);