C₅×#20263480 #44539189

order	31605
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#20263480

Splittings

Fitting, socle, Hall	C₅×C₇²×C₄₃ ⋊ C₃
derived, Hall	C₇²×C₄₃ ⋊ C₃×C₅
Hall	C₅×C₇² ⋊ C₄₃⋊C₃
Hall	C₅×C₄₃ ⋊ #3861
Sylow	C₇² ⋊ C₅×C₄₃⋊C₃
Sylow	C₄₃ ⋊ C₅×#3861

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#20263480
commutator, cosocle, hypocenter	C₇²×C₄₃	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₄₃	C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₅

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 62
transitive degree	μ_t(G) = 10535

A minimal faithful permutation representation — 3 generators on 62 points:

(1,2,3,4,5)(7,8,9)(10,12,11)(14,15,16)(17,19,18)(21,22,24)(23,26,31)(25,29,28)(27,33,41)(30,36,38)(32,39,35)(34,43,49)(37,45,50)(40,48,47)(42,51,44)(46,55,57)(52,58,56)(53,59,54)(60,62,61);(13,14,15,17,16,18,19)(20,21,23,27,34,36,22,25,30,37,46,45,26,32,40,49,57,55,33,42,52,59,41,50,43,53,60,62,51,31,38,47,56,61,58,39,24,28,35,44,54,48,29);(6,7,8,10,9,11,12)

Conjugacy classes & character table

3525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31605, 20);

C5×#20263480 #44539189