C₅×C₃₇×D₃₅ #27188317

order	12950
isoclinism	isoclinic class of D₃₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₃₇×D₃₅
unambiguous semidirect product	C₅⋊(C₅×D₇×C₃₇)
unambiguous semidirect product	C₇⋊(C₅×D₅×C₃₇)

Splittings

Fitting, socle, Hall	C₅²×C₇×C₃₇ ⋊ C₂
derived	C₅×C₇ ⋊ C₂×C₅×C₃₇
Hall	C₅²×C₃₇ ⋊ D₇
Hall	C₇×C₃₇ ⋊ C₅×D₅
Hall	C₅²×C₇ ⋊ C₂×C₃₇
Sylow	C₅² ⋊ D₇×C₃₇
Sylow	C₇ ⋊ C₅×D₅×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₃₇	D₃₅
commutator, cosocle, hypocenter	C₅×C₇	C₂×C₅×C₃₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₃₇	C₂
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₅×C₃₇

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 2590

A minimal faithful permutation representation — 2 generators on 54 points:

(2,3)(4,5)(6,7)(8,9,10,11,12)(14,15)(16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54);(1,2,4,6,7,5,3)(13,14,16,17,15)

Conjugacy classes & character table

3515 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₃₅×D₃₇, C₅×D₅×D₇×C₃₇, C₅×D₃₅⋊D₃₇, …
maximal normal subgroup of	C₅×D₃₅×D₃₇, C₅×D₅×D₇×C₃₇, C₅×D₃₅⋊D₃₇, …
maximal quotient of	C₅×C₃₇×Dic₃₅, C₂×C₅×C₃₇×D₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12950, 15);

C5×C37×D35 #27188317