Dic₈₅₈₈ #72272933

order	34352
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₅₈₈

Splittings

Hall	C₁₉×C₁₁₃ ⋊ Q₁₆
Sylow	C₁₁₃ ⋊ Dic₇₆
Sylow	C₁₉ ⋊ Dic₄₅₂

Subgroups

	subgroup	G/H
center	C₂	D₈₅₈₈
commutator, cosocle	C₄×C₁₉×C₁₁₃	C₂²
Frattini	C₄	C₂×D₂₁₄₇
Fitting	C₈×C₁₉×C₁₁₃	C₂
socle, prime radical	C₂×C₁₉×C₁₁₃	D₄
hypercenter	C₈	D₂₁₄₇
hypocenter	C₁₉×C₁₁₃	Q₁₆
LCS	G ▷ C₄×C₁₉×C₁₁₃ ▷ C₂×C₁₉×C₁₁₃ ▷ C₁₉×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 148
transitive degree	μ_t(G) = 34352

A minimal faithful permutation representation — 2 generators on 148 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(37,148)(38,147)(39,146)(40,145)(41,144)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)

Polycyclic presentation:

2^4,19,113|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^18,f^a=f^112

Conjugacy classes & character table

8591 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 19, 113]);;
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.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8588 #72272933