Dic₁₃₃₈₆ #72289162

order	53544
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₃₈₆

Splittings

Hall	C₃×C₂₃×C₉₇ ⋊ Q₈
Hall	C₂₃×C₉₇ ⋊ Dic₆
Hall	C₃×C₉₇ ⋊ Dic₄₆
Sylow	C₉₇ ⋊ Dic₁₃₈
Hall	C₃×C₂₃ ⋊ Dic₁₉₄
Sylow	C₂₃ ⋊ Dic₅₈₂
Sylow	C₃ ⋊ Dic₄₄₆₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₉₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₂₃×C₉₇	C₂²
Fitting	C₃×C₄×C₂₃×C₉₇	C₂
hypercenter	C₄	D₆₆₉₃
hypocenter	C₃×C₂₃×C₉₇	Q₈
LCS	G ▷ C₂×C₃×C₂₃×C₉₇ ▷ C₃×C₂₃×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 53544

A minimal faithful permutation representation — 2 generators on 131 points:

(1,2,3,4)(5,8,7,6)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(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);(1,5,3,7)(2,6,4,8)(10,11)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)

Polycyclic presentation:

2^3,3,23,97|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^22,f^a=f^96

Conjugacy classes & character table

13389 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, 3, 23, 97]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13386 #72289162