Dic₁₀₀₇₄ #72274170

order	40296
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₀₇₄

Splittings

Hall	C₃×C₂₃×C₇₃ ⋊ Q₈
Hall	C₂₃×C₇₃ ⋊ Dic₆
Hall	C₃×C₇₃ ⋊ Dic₄₆
Sylow	C₇₃ ⋊ Dic₁₃₈
Hall	C₃×C₂₃ ⋊ Dic₁₄₆
Sylow	C₂₃ ⋊ Dic₄₃₈
Sylow	C₃ ⋊ Dic₃₃₅₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₀₃₇
commutator, socle, prime radical, cosocle	C₂×C₃×C₂₃×C₇₃	C₂²
Fitting	C₃×C₄×C₂₃×C₇₃	C₂
hypercenter	C₄	D₅₀₃₇
hypocenter	C₃×C₂₃×C₇₃	Q₈
LCS	G ▷ C₂×C₃×C₂₃×C₇₃ ▷ C₃×C₂₃×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 40296

A minimal faithful permutation representation — 2 generators on 107 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);(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)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^3,3,23,73|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^22,f^a=f^72

Conjugacy classes & character table

10077 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, 73]);;
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.6^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10074 #72274170