D₂₇₉₂₄ #72291241

order	55848
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₉₂₄

Splittings

Fitting	C₃×C₄×C₁₃×C₁₇₉ ⋊ C₂
Hall	C₃×C₁₃×C₁₇₉ ⋊ D₄
Hall	C₁₃×C₁₇₉ ⋊ D₁₂
Hall	C₃×C₁₇₉ ⋊ D₅₂
Sylow	C₁₇₉ ⋊ D₁₅₆
Hall	C₃×C₁₃ ⋊ D₇₁₆
Sylow	C₁₃ ⋊ D₂₁₄₈
Sylow	C₃ ⋊ D₉₃₀₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₉₈₁
commutator, socle, cosocle	C₂×C₃×C₁₃×C₁₇₉	C₂²
Fitting	C₃×C₄×C₁₃×C₁₇₉	C₂
hypercenter	C₄	D₆₉₈₁
hypocenter	C₃×C₁₃×C₁₇₉	D₄
LCS	G ▷ C₂×C₃×C₁₃×C₁₇₉ ▷ C₃×C₁₃×C₁₇₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 199
transitive degree	μ_t(G) = 27924

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

Polycyclic presentation:

2^3,3,13,179|b^2=c,b^a=bc,d^a=d^2,e^a=e^12,f^a=f^178

Conjugacy classes & character table

13965 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, 2, 2, 3, 13, 179]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^178);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27924 #72291241