Dic₃₈₁₆ #72215065

order	15264
automorphism	Aut(G) order 19049472
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₈₁₆

Splittings

Hall	C₉×C₅₃ ⋊ Q₃₂
Sylow	C₅₃ ⋊ Dic₇₂
Sylow	C₉ ⋊ Dic₄₂₄

Subgroups

	subgroup	G/H
center	C₂	D₃₈₁₆
commutator, cosocle	C₈×C₉×C₅₃	C₂²
Frattini	C₃×C₈	C₂×D₁₅₉
Fitting	C₉×C₁₆×C₅₃	C₂
socle, prime radical	C₂×C₃×C₅₃	D₂₄
hypercenter	C₁₆	D₄₇₇
hypocenter	C₉×C₅₃	Q₃₂
LCS	G ▷ C₈×C₉×C₅₃ ▷ C₄×C₉×C₅₃ ▷ C₂×C₉×C₅₃ ▷ C₉×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 94
transitive degree	μ_t(G) = 15264

A minimal faithful permutation representation — 2 generators on 94 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,41)(35,40)(36,39)(37,38)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^5,3,53,3|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=h,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2,g^a=g^52,h^a=h^2

Conjugacy classes & character table

3819 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₆₃₂, Dic₁₁₄₄₈
maximal normal subgroup of	Dic₇₆₃₂
maximal quotient of	Dic₁₁₄₄₈

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 53, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^52);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3816 #72215065