Dic₇₉₅₈ #72272385

order	31832
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₉₅₈

Splittings

Hall	C₂₃×C₁₇₃ ⋊ Q₈
Sylow	C₁₇₃ ⋊ Dic₄₆
Sylow	C₂₃ ⋊ Dic₃₄₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₉₇₉
commutator, socle, prime radical, cosocle	C₂×C₂₃×C₁₇₃	C₂²
Fitting	C₄×C₂₃×C₁₇₃	C₂
hypercenter	C₄	D₃₉₇₉
hypocenter	C₂₃×C₁₇₃	Q₈
LCS	G ▷ C₂×C₂₃×C₁₇₃ ▷ C₂₃×C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 204
transitive degree	μ_t(G) = 31832

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

Polycyclic presentation:

2^3,23,173|a^2=c,b^2=c,b^a=bc,d^a=d^22,e^a=e^172

Conjugacy classes & character table

7961 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₉₁₆
maximal normal subgroup of	Dic₁₅₉₁₆

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, 23, 173]);;
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.4^22);;
gap> SetConjugate(coll, 5, 1, F.5^172);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7958 #72272385