Dic₁₁₂₂₄ #72275307

order	44896
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₂₂₄

Splittings

Hall	C₂₃×C₆₁ ⋊ Q₃₂
Sylow	C₆₁ ⋊ Dic₁₈₄
Sylow	C₂₃ ⋊ Dic₄₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₁₂₂₄
commutator, cosocle	C₈×C₂₃×C₆₁	C₂²
Frattini	C₈	C₂×D₁₄₀₃
Fitting	C₁₆×C₂₃×C₆₁	C₂
socle, prime radical	C₂×C₂₃×C₆₁	D₈
hypercenter	C₁₆	D₁₄₀₃
hypocenter	C₂₃×C₆₁	Q₃₂
LCS	G ▷ C₈×C₂₃×C₆₁ ▷ C₄×C₂₃×C₆₁ ▷ C₂×C₂₃×C₆₁ ▷ C₂₃×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 44896

A minimal faithful permutation representation — 2 generators on 116 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2^5,23,61|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^22,g^a=g^60

Conjugacy classes & character table

11227 conjugacy classes. The character table is too large to display.

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 23, 61]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11224 #72275307