D₂₈₅₂₅ #72292149

order	57050
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₈₅₂₅

Splittings

Fitting, derived, Hall	C₇×C₂₅×C₁₆₃ ⋊ C₂
Hall	C₂₅×C₁₆₃ ⋊ D₇
Hall	C₇×C₁₆₃ ⋊ D₂₅
Hall	C₇×C₂₅ ⋊ D₁₆₃
Sylow	C₁₆₃ ⋊ D₁₇₅
Sylow	C₂₅ ⋊ D₁₁₄₁
Sylow	C₇ ⋊ D₄₀₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₅×C₁₆₃	C₂
Frattini	C₅	D₅₇₀₅
socle	C₅×C₇×C₁₆₃	D₅
LCS	G ▷ C₇×C₂₅×C₁₆₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 195
transitive degree	μ_t(G) = 28525

A minimal faithful permutation representation — 2 generators on 195 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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,32)(28,31)(29,30)(34,195)(35,194)(36,193)(37,192)(38,191)(39,190)(40,189)(41,188)(42,187)(43,186)(44,185)(45,184)(46,183)(47,182)(48,181)(49,180)(50,179)(51,178)(52,177)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,169)(61,168)(62,167)(63,166)(64,165)(65,164)(66,163)(67,162)(68,161)(69,160)(70,159)(71,158)(72,157)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(81,148)(82,147)(83,146)(84,145)(85,144)(86,143)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)

Polycyclic presentation:

2,5,7,163,5|b^5=e,b^a=b^4e^4,c^a=c^6,d^a=d^162,e^a=e^4

Conjugacy classes & character table

14264 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 5, 7, 163, 5]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^162);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28525 #72292149