Dic₄₃₆₀ #72215801

order	17440
automorphism	Aut(G) order 30136320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₆₀

Splittings

Hall	C₅×C₁₀₉ ⋊ Q₃₂
Sylow	C₁₀₉ ⋊ Dic₄₀
Sylow	C₅ ⋊ Dic₈₇₂

Subgroups

	subgroup	G/H
center	C₂	D₄₃₆₀
commutator, cosocle	C₅×C₈×C₁₀₉	C₂²
Frattini	C₈	C₂×D₅₄₅
Fitting	C₅×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂×C₅×C₁₀₉	D₈
hypercenter	C₁₆	D₅₄₅
hypocenter	C₅×C₁₀₉	Q₃₂
LCS	G ▷ C₅×C₈×C₁₀₉ ▷ C₄×C₅×C₁₀₉ ▷ C₂×C₅×C₁₀₉ ▷ C₅×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 146
transitive degree	μ_t(G) = 17440

A minimal faithful permutation representation — 2 generators on 146 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146);(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)(34,37)(35,36)(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:

2^5,5,109|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^108

Conjugacy classes & character table

4363 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₂₀, Dic₁₃₀₈₀
maximal normal subgroup of	Dic₈₇₂₀
maximal quotient of	Dic₁₃₀₈₀

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 109]);;
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> 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.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4360 #72215801