Dic₆₉₆₈ #72236366

order	27872
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₉₆₈

Splittings

Hall	C₁₃×C₆₇ ⋊ Q₃₂
Sylow	C₆₇ ⋊ Dic₁₀₄
Sylow	C₁₃ ⋊ Dic₅₃₆

Subgroups

	subgroup	G/H
center	C₂	D₆₉₆₈
commutator, cosocle	C₈×C₁₃×C₆₇	C₂²
Frattini	C₈	C₂×D₈₇₁
Fitting	C₁₃×C₁₆×C₆₇	C₂
socle, prime radical	C₂×C₁₃×C₆₇	D₈
hypercenter	C₁₆	D₈₇₁
hypocenter	C₁₃×C₆₇	Q₃₂
LCS	G ▷ C₈×C₁₃×C₆₇ ▷ C₄×C₁₃×C₆₇ ▷ C₂×C₁₃×C₆₇ ▷ C₁₃×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 27872

A minimal faithful permutation representation — 2 generators on 112 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);(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)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)

Polycyclic presentation:

2^5,13,67|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^12,g^a=g^66

Conjugacy classes & character table

6971 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₉₃₆
maximal normal subgroup of	Dic₁₃₉₃₆

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, 13, 67]);;
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.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6968 #72236366