C₅×#20299799 #49136404

order	36260
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₇

Aliases

direct product

C₅×#20299799

Splittings

Fitting, socle, Hall	C₅×C₇²×C₃₇ ⋊ C₄
derived, Hall	C₇²×C₃₇ ⋊ C₄×C₅
Hall	C₅×C₇² ⋊ #3866
Hall	C₅×C₃₇ ⋊ #5997
Sylow	C₇² ⋊ C₅×#3866
Sylow	C₃₇ ⋊ C₅×#5997

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#20299799
commutator, hypocenter	C₇²×C₃₇	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₃₇	C₄
prime radical	C₅×C₇⋊D₂₅₉	C₂
cosocle	C₇⋊D₂₅₉	C₂×C₅
LCS	G ▷ C₇²×C₃₇
UCS	C₁ ◁ C₅

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 2590

A minimal faithful permutation representation — 2 generators on 56 points:

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

Conjugacy classes & character table

2285 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36260, 50);

C5×#20299799 #49136404