Dic₁₅₄₂₃ #72293512

order	61692
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₄₂₃

Splittings

derived, Hall	C₃×C₅₃×C₉₇ ⋊ C₄
Hall	C₅₃×C₉₇ ⋊ Dic₃
Hall	C₃×C₉₇ ⋊ Dic₅₃
Hall	C₃×C₅₃ ⋊ Dic₉₇
Sylow	C₉₇ ⋊ Dic₁₅₉
Sylow	C₅₃ ⋊ Dic₂₉₁
Sylow	C₃ ⋊ Dic₅₁₄₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₄₂₃
commutator, hypocenter	C₃×C₅₃×C₉₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅₃×C₉₇	C₂
LCS	G ▷ C₃×C₅₃×C₉₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 61692

A minimal faithful permutation representation — 2 generators on 157 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);(1,3,2,4)(6,7)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,140)(80,139)(81,138)(82,137)(83,136)(84,135)(85,134)(86,133)(87,132)(88,131)(89,130)(90,129)(91,128)(92,127)(93,126)(94,125)(95,124)(96,123)(97,122)(98,121)(99,120)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)

Polycyclic presentation:

2^2,3,53,97|a^2=b,c^a=c^2,d^a=d^52,e^a=e^96

Conjugacy classes & character table

15426 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, 2, 3, 53, 97]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^52);;
gap> SetConjugate(coll, 5, 1, F.5^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15423 #72293512