Dic₂₃₆₄ #72212698

order	9456
automorphism	Aut(G) order 7413504
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₃₆₄

Splittings

Hall	C₃×C₁₉₇ ⋊ Q₁₆
Sylow	C₁₉₇ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₇₈₈

Subgroups

	subgroup	G/H
center	C₂	D₂₃₆₄
commutator, cosocle	C₃×C₄×C₁₉₇	C₂²
Frattini	C₄	C₂×D₅₉₁
Fitting	C₃×C₈×C₁₉₇	C₂
socle, prime radical	C₂×C₃×C₁₉₇	D₄
hypercenter	C₈	D₅₉₁
hypocenter	C₃×C₁₉₇	Q₁₆
LCS	G ▷ C₃×C₄×C₁₉₇ ▷ C₂×C₃×C₁₉₇ ▷ C₃×C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 216
transitive degree	μ_t(G) = 9456

A minimal faithful permutation representation — 2 generators on 216 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,216)(22,215)(23,214)(24,213)(25,212)(26,211)(27,210)(28,209)(29,208)(30,207)(31,206)(32,205)(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:

2^4,3,197|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^196

Conjugacy classes & character table

2367 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₇₂₈, Dic₇₀₉₂, Dic₁₁₈₂₀
maximal normal subgroup of	Dic₄₇₂₈
maximal quotient of	Dic₇₀₉₂, Dic₁₁₈₂₀

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, 197]);;
gap> SetPower(coll, 1, F.4);;
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.6^196);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2364 #72212698