D₅₃₅₂ #72211653

order	10704
automorphism	Aut(G) order 9505152
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₅₃₅₂

Splittings

Fitting	C₃×C₈×C₂₂₃ ⋊ C₂
Hall	C₃×C₂₂₃ ⋊ D₈
Sylow	C₂₂₃ ⋊ D₂₄
Sylow	C₃ ⋊ D₁₇₈₄

Subgroups

	subgroup	G/H
center	C₂	D₂₆₇₆
commutator, cosocle	C₃×C₄×C₂₂₃	C₂²
Frattini	C₄	C₂×D₆₆₉
Fitting	C₃×C₈×C₂₂₃	C₂
socle	C₂×C₃×C₂₂₃	D₄
hypercenter	C₈	D₆₆₉
hypocenter	C₃×C₂₂₃	D₈
LCS	G ▷ C₃×C₄×C₂₂₃ ▷ C₂×C₃×C₂₂₃ ▷ C₃×C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 234
transitive degree	μ_t(G) = 5352

A minimal faithful permutation representation — 2 generators on 234 points:

(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234);(2,8)(3,7)(4,6)(10,11)(13,234)(14,233)(15,232)(16,231)(17,230)(18,229)(19,228)(20,227)(21,226)(22,225)(23,224)(24,223)(25,222)(26,221)(27,220)(28,219)(29,218)(30,217)(31,216)(32,215)(33,214)(34,213)(35,212)(36,211)(37,210)(38,209)(39,208)(40,207)(41,206)(42,205)(43,204)(44,203)(45,202)(46,201)(47,200)(48,199)(49,198)(50,197)(51,196)(52,195)(53,194)(54,193)(55,192)(56,191)(57,190)(58,189)(59,188)(60,187)(61,186)(62,185)(63,184)(64,183)(65,182)(66,181)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)

Polycyclic presentation:

2^4,3,223|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^222

Conjugacy classes & character table

2679 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₀₇₀₄, D₁₆₀₅₆, D₂₆₇₆₀
maximal normal subgroup of	D₁₀₇₀₄
maximal quotient of	D₁₀₇₀₄, Dic₅₃₅₂, D₁₆₀₅₆, …

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, 2, 3, 223]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D5352 #72211653