C₅×#19096903 #37590397

order	24660
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₅×#19096903

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₃₇ ⋊ C₄
derived, Hall	C₃²×C₁₃₇ ⋊ C₄×C₅
Hall	C₅×C₁₃₇ ⋊ #157
Sylow	C₁₃₇ ⋊ C₅×#157
Hall	C₃²×C₅ ⋊ #10587739
Sylow	C₃² ⋊ C₅×#10587739

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#19096903
commutator, hypocenter	C₃²×C₁₃₇	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃₇	C₄
prime radical	C₅×C₃⋊D₄₁₁	C₂
cosocle	C₃⋊D₄₁₁	C₂×C₅
LCS	G ▷ C₃²×C₁₃₇
UCS	C₁ ◁ C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 148
transitive degree	μ_t(G) = 4110

A minimal faithful permutation representation — 2 generators on 148 points:

(1,2,3,4,5)(7,8,9,11)(10,13,14,20)(12,17,18,16)(15,22,23,33)(19,26,27,29)(21,30,31,25)(24,35,36,50)(28,39,40,46)(32,43,44,42)(34,47,48,38)(37,52,53,71)(41,56,57,67)(45,60,61,63)(49,64,65,59)(51,68,69,55)(54,73,74,96)(58,77,78,92)(62,81,82,88)(66,85,86,84)(70,89,90,80)(72,93,94,76)(75,98,99,120)(79,102,103,116)(83,106,107,112)(87,109,104,110)(91,111,100,108)(95,113,114,105)(97,117,118,101)(115,125,126,124)(119,128,129,123)(121,127,131,122)(130,135,136,134)(132,137,138,133)(139,141,142,140)(143,144)(145,146,147,148);(6,7,10,15,24,37,54,75,100,90,69,50,67,88,110,98,121,132,139,141,135,125,111,85,60,39,22,34,49,66,87,103,78,57,40,27,18,11,16,25,38,55,76,101,122,126,114,94,71,92,112,102,73,97,119,130,137,128,113,89,64,43,26,13,21,32,45,62,83,107,82,61,44,31,20,29,42,59,80,105,123,133,136,129,118,96,116,106,77,52,72,95,115,127,117,93,68,47,30,17,8,12,19,28,41,58,79,104,86,65,48,33,46,63,84,108,124,134,140,142,138,131,120,109,81,56,35,51,70,91,99,74,53,36,23,14,9)(143,145,147)(144,148,146)

Conjugacy classes & character table

1560 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24660, 51);

C5×#19096903 #37590397