Dic₈₁₈₈ #72272575

order	32752
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₁₈₈

Splittings

Hall	C₂₃×C₈₉ ⋊ Q₁₆
Sylow	C₈₉ ⋊ Dic₉₂
Sylow	C₂₃ ⋊ Dic₃₅₆

Subgroups

	subgroup	G/H
center	C₂	D₈₁₈₈
commutator, cosocle	C₄×C₂₃×C₈₉	C₂²
Frattini	C₄	C₂×D₂₀₄₇
Fitting	C₈×C₂₃×C₈₉	C₂
socle, prime radical	C₂×C₂₃×C₈₉	D₄
hypercenter	C₈	D₂₀₄₇
hypocenter	C₂₃×C₈₉	Q₁₆
LCS	G ▷ C₄×C₂₃×C₈₉ ▷ C₂×C₂₃×C₈₉ ▷ C₂₃×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 32752

A minimal faithful permutation representation — 2 generators on 128 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)

Polycyclic presentation:

2^4,23,89|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^88

Conjugacy classes & character table

8191 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₃₇₆
maximal normal subgroup of	Dic₁₆₃₇₆

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, 23, 89]);;
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.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8188 #72272575