D₁₂₄₈₅ #38735838

order	24970
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₄₈₅
unambiguous semidirect product	C₅⋊D₂₄₉₇
unambiguous semidirect product	C₁₁⋊D₁₁₃₅
unambiguous semidirect product	C₂₂₇⋊D₅₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₁×C₂₂₇ ⋊ C₂
Hall	C₁₁×C₂₂₇ ⋊ D₅
Hall	C₅×C₂₂₇ ⋊ D₁₁
Sylow	C₂₂₇ ⋊ D₅₅
Hall	C₅×C₁₁ ⋊ D₂₂₇
Sylow	C₁₁ ⋊ D₁₁₃₅
Sylow	C₅ ⋊ D₂₄₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₁×C₂₂₇	C₂
LCS	G ▷ C₅×C₁₁×C₂₂₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 243
transitive degree	μ_t(G) = 12485

A minimal faithful permutation representation — 2 generators on 243 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243);(2,5)(3,4)(7,16)(8,15)(9,14)(10,13)(11,12)(18,243)(19,242)(20,241)(21,240)(22,239)(23,238)(24,237)(25,236)(26,235)(27,234)(28,233)(29,232)(30,231)(31,230)(32,229)(33,228)(34,227)(35,226)(36,225)(37,224)(38,223)(39,222)(40,221)(41,220)(42,219)(43,218)(44,217)(45,216)(46,215)(47,214)(48,213)(49,212)(50,211)(51,210)(52,209)(53,208)(54,207)(55,206)(56,205)(57,204)(58,203)(59,202)(60,201)(61,200)(62,199)(63,198)(64,197)(65,196)(66,195)(67,194)(68,193)(69,192)(70,191)(71,190)(72,189)(73,188)(74,187)(75,186)(76,185)(77,184)(78,183)(79,182)(80,181)(81,180)(82,179)(83,178)(84,177)(85,176)(86,175)(87,174)(88,173)(89,172)(90,171)(91,170)(92,169)(93,168)(94,167)(95,166)(96,165)(97,164)(98,163)(99,162)(100,161)(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)

Polycyclic presentation:

2,5,11,227|b^a=b^4,c^a=c^10,d^a=d^226

Conjugacy classes & character table

6244 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24970, 9);

D12485 #38735838