Dic₁₅₁₃₂ #72292599

order	60528
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₁₃₂

Splittings

Hall	C₃×C₁₃×C₉₇ ⋊ Q₁₆
Hall	C₁₃×C₉₇ ⋊ Dic₁₂
Hall	C₃×C₉₇ ⋊ Dic₅₂
Sylow	C₉₇ ⋊ Dic₁₅₆
Hall	C₃×C₁₃ ⋊ Dic₃₈₈
Sylow	C₁₃ ⋊ Dic₁₁₆₄
Sylow	C₃ ⋊ Dic₅₀₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₅₁₃₂
commutator, cosocle	C₃×C₄×C₁₃×C₉₇	C₂²
Frattini	C₄	C₂×D₃₇₈₃
Fitting	C₃×C₈×C₁₃×C₉₇	C₂
socle, prime radical	C₂×C₃×C₁₃×C₉₇	D₄
hypercenter	C₈	D₃₇₈₃
hypocenter	C₃×C₁₃×C₉₇	Q₁₆
LCS	G ▷ C₃×C₄×C₁₃×C₉₇ ▷ C₂×C₃×C₁₃×C₉₇ ▷ C₃×C₁₃×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 60528

A minimal faithful permutation representation — 2 generators on 129 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^4,3,13,97|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^12,g^a=g^96

Conjugacy classes & character table

15135 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, 3, 13, 97]);;
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.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15132 #72292599