Dic₅₅₄₈ #72217701

order	22192
automorphism	Aut(G) order 57521664
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₅₄₈

Splittings

Hall	C₁₉×C₇₃ ⋊ Q₁₆
Sylow	C₇₃ ⋊ Dic₇₆
Sylow	C₁₉ ⋊ Dic₂₉₂

Subgroups

	subgroup	G/H
center	C₂	D₅₅₄₈
commutator, cosocle	C₄×C₁₉×C₇₃	C₂²
Frattini	C₄	C₂×D₁₃₈₇
Fitting	C₈×C₁₉×C₇₃	C₂
socle, prime radical	C₂×C₁₉×C₇₃	D₄
hypercenter	C₈	D₁₃₈₇
hypocenter	C₁₉×C₇₃	Q₁₆
LCS	G ▷ C₄×C₁₉×C₇₃ ▷ C₂×C₁₉×C₇₃ ▷ C₁₉×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 22192

A minimal faithful permutation representation — 2 generators on 108 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);(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)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2^4,19,73|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^18,f^a=f^72

Conjugacy classes & character table

5551 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₀₉₆
maximal normal subgroup of	Dic₁₁₀₉₆

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

Dic5548 #72217701