Dic₂₇₄₇ #24926913

order	10988
automorphism	Aut(G) order 14504160
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₄₇
unambiguous semidirect product	C₆₇⋊Dic₄₁

Splittings

derived, Hall	C₄₁×C₆₇ ⋊ C₄
Sylow	C₆₇ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₇₄₇
commutator, hypocenter	C₄₁×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₆₇	C₂
LCS	G ▷ C₄₁×C₆₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 10988

A minimal faithful permutation representation — 2 generators on 112 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112);(1,3,2,4)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(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:

2,41,67,2|a^2=d,b^a=b^40,c^a=c^66

Conjugacy classes & character table

2750 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₇₄₇, Dic₈₂₄₁, (C₆₇⋊C₃)⋊Dic₄₁, …
maximal normal subgroup of	C₃×Dic₂₇₄₇, (C₆₇⋊C₃)⋊Dic₄₁, Dic₅₄₉₄
maximal quotient of	C₃×Dic₂₇₄₇, Dic₈₂₄₁, Dic₁₃₇₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10988, 4);

Dic2747 #24926913