D₁₁₅₀₈ #72223908

order	23016
automorphism	Aut(G) order 37562112
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₅₀₈

Splittings

Fitting	C₃×C₄×C₇×C₁₃₇ ⋊ C₂
Hall	C₃×C₇×C₁₃₇ ⋊ D₄
Hall	C₇×C₁₃₇ ⋊ D₁₂
Hall	C₃×C₁₃₇ ⋊ D₂₈
Sylow	C₁₃₇ ⋊ D₈₄
Hall	C₃×C₇ ⋊ D₅₄₈
Sylow	C₇ ⋊ D₁₆₄₄
Sylow	C₃ ⋊ D₃₈₃₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₈₇₇
commutator, socle, cosocle	C₂×C₃×C₇×C₁₃₇	C₂²
Fitting	C₃×C₄×C₇×C₁₃₇	C₂
hypercenter	C₄	D₂₈₇₇
hypocenter	C₃×C₇×C₁₃₇	D₄
LCS	G ▷ C₂×C₃×C₇×C₁₃₇ ▷ C₃×C₇×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 151
transitive degree	μ_t(G) = 11508

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

Polycyclic presentation:

2^3,3,7,137|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^136

Conjugacy classes & character table

5757 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₀₁₆
maximal normal subgroup of	D₂₃₀₁₆
maximal quotient of	Dic₁₁₅₀₈, D₂₃₀₁₆

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, 137]);;
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.6^136);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11508 #72223908