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C₇×D₃₆ #92605

order	504
isoclinism	isoclinic class of Dic₁₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×D₃₆
unambiguous semidirect product	C₄⋊(C₇×D₉)

Splittings

Fitting	C₄×C₇×C₉ ⋊ C₂
Hall	C₇×C₉ ⋊ D₄
Sylow	C₉ ⋊ C₇×D₄

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂×D₉
commutator, cosocle	C₂×C₉	C₂²×C₇
Frattini	C₂×C₃	C₂×D₃×C₇
Fitting	C₄×C₇×C₉	C₂
socle	C₂×C₃×C₇	C₂×D₃
hypercenter	C₄×C₇	D₉
hypocenter	C₉	C₇×D₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 252

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11758856, #11758858, #11758862, …
maximal normal subgroup of	#11758856, #11758858, #11758862, …
maximal quotient of	C₇×Dic₃₆, C₇×#3664, C₇×D₇₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 31);