Dic₃₁₂₂ #72213413

order	12488
automorphism	Aut(G) order 16634016
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₁₂₂

Splittings

Hall	C₇×C₂₂₃ ⋊ Q₈
Sylow	C₂₂₃ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₄₄₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₅₆₁
commutator, socle, prime radical, cosocle	C₂×C₇×C₂₂₃	C₂²
Fitting	C₄×C₇×C₂₂₃	C₂
hypercenter	C₄	D₁₅₆₁
hypocenter	C₇×C₂₂₃	Q₈
LCS	G ▷ C₂×C₇×C₂₂₃ ▷ C₇×C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 238
transitive degree	μ_t(G) = 12488

A minimal faithful permutation representation — 2 generators on 238 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,238)(18,237)(19,236)(20,235)(21,234)(22,233)(23,232)(24,231)(25,230)(26,229)(27,228)(28,227)(29,226)(30,225)(31,224)(32,223)(33,222)(34,221)(35,220)(36,219)(37,218)(38,217)(39,216)(40,215)(41,214)(42,213)(43,212)(44,211)(45,210)(46,209)(47,208)(48,207)(49,206)(50,205)(51,204)(52,203)(53,202)(54,201)(55,200)(56,199)(57,198)(58,197)(59,196)(60,195)(61,194)(62,193)(63,192)(64,191)(65,190)(66,189)(67,188)(68,187)(69,186)(70,185)(71,184)(72,183)(73,182)(74,181)(75,180)(76,179)(77,178)(78,177)(79,176)(80,175)(81,174)(82,173)(83,172)(84,171)(85,170)(86,169)(87,168)(88,167)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)

Polycyclic presentation:

2^3,7,223|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^222

Conjugacy classes & character table

3125 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₂₄₄, Dic₉₃₆₆, Dic₁₅₆₁₀
maximal normal subgroup of	Dic₆₂₄₄
maximal quotient of	Dic₉₃₆₆, Dic₁₅₆₁₀

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, 7, 223]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3122 #72213413