Dic₇₄₁₆ #72236613

order	29664
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₄₁₆

Splittings

Hall	C₉×C₁₀₃ ⋊ Q₃₂
Sylow	C₁₀₃ ⋊ Dic₇₂
Sylow	C₉ ⋊ Dic₈₂₄

Subgroups

	subgroup	G/H
center	C₂	D₇₄₁₆
commutator, cosocle	C₈×C₉×C₁₀₃	C₂²
Frattini	C₃×C₈	C₂×D₃₀₉
Fitting	C₉×C₁₆×C₁₀₃	C₂
socle, prime radical	C₂×C₃×C₁₀₃	D₂₄
hypercenter	C₁₆	D₉₂₇
hypocenter	C₉×C₁₀₃	Q₃₂
LCS	G ▷ C₈×C₉×C₁₀₃ ▷ C₄×C₉×C₁₀₃ ▷ C₂×C₉×C₁₀₃ ▷ C₉×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 29664

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

Polycyclic presentation:

2^5,3,103,3|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^102,h^a=h^2

Conjugacy classes & character table

7419 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₈₃₂
maximal normal subgroup of	Dic₁₄₈₃₂

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, 103, 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.7^102);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7416 #72236613