D₂₂₅₄₅ #72277192

order	45090
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₂₅₄₅

Splittings

Fitting, derived, Hall	C₅×C₂₇×C₁₆₇ ⋊ C₂
Hall	C₂₇×C₁₆₇ ⋊ D₅
Hall	C₅×C₁₆₇ ⋊ D₂₇
Sylow	C₁₆₇ ⋊ D₁₃₅
Hall	C₅×C₂₇ ⋊ D₁₆₇
Sylow	C₂₇ ⋊ D₈₃₅
Sylow	C₅ ⋊ D₄₅₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₂₇×C₁₆₇	C₂
Frattini	C₉	D₂₅₀₅
socle	C₃×C₅×C₁₆₇	D₉
LCS	G ▷ C₅×C₂₇×C₁₆₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 199
transitive degree	μ_t(G) = 22545

A minimal faithful permutation representation — 2 generators on 199 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(34,199)(35,198)(36,197)(37,196)(38,195)(39,194)(40,193)(41,192)(42,191)(43,190)(44,189)(45,188)(46,187)(47,186)(48,185)(49,184)(50,183)(51,182)(52,181)(53,180)(54,179)(55,178)(56,177)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,160)(74,159)(75,158)(76,157)(77,156)(78,155)(79,154)(80,153)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,117)

Polycyclic presentation:

2,3,5,167,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^4,d^a=d^166,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

11274 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", "f");;
gap> coll := SingleCollector(F, [2, 3, 5, 167, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^166);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D22545 #72277192