D₁₁₃₅₆ #72223893

order	22712
automorphism	Aut(G) order 60323072
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₃₅₆

Splittings

Fitting	C₄×C₁₇×C₁₆₇ ⋊ C₂
Hall	C₁₇×C₁₆₇ ⋊ D₄
Sylow	C₁₆₇ ⋊ D₆₈
Sylow	C₁₇ ⋊ D₆₆₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₈₃₉
commutator, socle, cosocle	C₂×C₁₇×C₁₆₇	C₂²
Fitting	C₄×C₁₇×C₁₆₇	C₂
hypercenter	C₄	D₂₈₃₉
hypocenter	C₁₇×C₁₆₇	D₄
LCS	G ▷ C₂×C₁₇×C₁₆₇ ▷ C₁₇×C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 188
transitive degree	μ_t(G) = 11356

A minimal faithful permutation representation — 2 generators on 188 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);(2,4)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(23,188)(24,187)(25,186)(26,185)(27,184)(28,183)(29,182)(30,181)(31,180)(32,179)(33,178)(34,177)(35,176)(36,175)(37,174)(38,173)(39,172)(40,171)(41,170)(42,169)(43,168)(44,167)(45,166)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,154)(58,153)(59,152)(60,151)(61,150)(62,149)(63,148)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)

Polycyclic presentation:

2^3,17,167|b^2=c,b^a=bc,d^a=d^16,e^a=e^166

Conjugacy classes & character table

5681 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₇₁₂
maximal normal subgroup of	D₂₂₇₁₂
maximal quotient of	Dic₁₁₃₅₆, D₂₂₇₁₂

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, 2, 17, 167]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^166);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11356 #72223893