C₃×D₄₁₈₇ #38737486

order	25122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₄₁₈₇
unambiguous semidirect product	C₅₃⋊(C₃×D₇₉)

Splittings

Fitting, socle, Hall	C₃×C₅₃×C₇₉ ⋊ C₂
derived, Hall	C₅₃×C₇₉ ⋊ C₂×C₃
Hall	C₃×C₇₉ ⋊ D₅₃
Hall	C₃×C₅₃ ⋊ D₇₉
Sylow	C₇₉ ⋊ C₃×D₅₃
Sylow	C₅₃ ⋊ C₃×D₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₄₁₈₇
commutator, cosocle, hypocenter	C₅₃×C₇₉	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅₃×C₇₉	C₂
LCS	G ▷ C₅₃×C₇₉
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 12561

A minimal faithful permutation representation — 2 generators on 135 points:

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

Conjugacy classes & character table

6285 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25122, 5);

C3×D4187 #38737486