D₂₆₇₆ #72210586

order	5352
automorphism	Aut(G) order 2376288
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₇₆

Splittings

Fitting	C₃×C₄×C₂₂₃ ⋊ C₂
Hall	C₃×C₂₂₃ ⋊ D₄
Sylow	C₂₂₃ ⋊ D₁₂
Sylow	C₃ ⋊ D₈₉₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₉
commutator, socle, cosocle	C₂×C₃×C₂₂₃	C₂²
Fitting	C₃×C₄×C₂₂₃	C₂
hypercenter	C₄	D₆₆₉
hypocenter	C₃×C₂₂₃	D₄
LCS	G ▷ C₂×C₃×C₂₂₃ ▷ C₃×C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 230
transitive degree	μ_t(G) = 2676

A minimal faithful permutation representation — 2 generators on 230 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230);(2,4)(6,7)(9,230)(10,229)(11,228)(12,227)(13,226)(14,225)(15,224)(16,223)(17,222)(18,221)(19,220)(20,219)(21,218)(22,217)(23,216)(24,215)(25,214)(26,213)(27,212)(28,211)(29,210)(30,209)(31,208)(32,207)(33,206)(34,205)(35,204)(36,203)(37,202)(38,201)(39,200)(40,199)(41,198)(42,197)(43,196)(44,195)(45,194)(46,193)(47,192)(48,191)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,164)(76,163)(77,162)(78,161)(79,160)(80,159)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)

Polycyclic presentation:

2^3,3,223|b^2=c,b^a=bc,d^a=d^2,e^a=e^222

Conjugacy classes & character table

1341 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃₅₂, D₈₀₂₈, D₁₃₃₈₀, …
maximal normal subgroup of	D₅₃₅₂
maximal quotient of	D₅₃₅₂, Dic₂₆₇₆, D₈₀₂₈, …

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, 3, 223]);;
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.5^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2676 #72210586