D₁₁₆₇₆ #72223935

order	23352
automorphism	Aut(G) order 38670912
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₆₇₆

Splittings

Fitting	C₃×C₄×C₇×C₁₃₉ ⋊ C₂
Hall	C₃×C₇×C₁₃₉ ⋊ D₄
Hall	C₇×C₁₃₉ ⋊ D₁₂
Hall	C₃×C₁₃₉ ⋊ D₂₈
Sylow	C₁₃₉ ⋊ D₈₄
Hall	C₃×C₇ ⋊ D₅₅₆
Sylow	C₇ ⋊ D₁₆₆₈
Sylow	C₃ ⋊ D₃₈₉₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₁₉
commutator, socle, cosocle	C₂×C₃×C₇×C₁₃₉	C₂²
Fitting	C₃×C₄×C₇×C₁₃₉	C₂
hypercenter	C₄	D₂₉₁₉
hypocenter	C₃×C₇×C₁₃₉	D₄
LCS	G ▷ C₂×C₃×C₇×C₁₃₉ ▷ C₃×C₇×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 153
transitive degree	μ_t(G) = 11676

A minimal faithful permutation representation — 2 generators on 153 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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)

Polycyclic presentation:

2^3,3,7,139|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^138

Conjugacy classes & character table

5841 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₃₅₂
maximal normal subgroup of	D₂₃₃₅₂
maximal quotient of	Dic₁₁₆₇₆, D₂₃₃₅₂

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, 7, 139]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11676 #72223935