Dic₃₃₃₇ #27211150

order	13348
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₃₃₇
unambiguous semidirect product	C₄₇⋊Dic₇₁
unambiguous semidirect product	C₇₁⋊Dic₄₇

Splittings

derived, Hall	C₄₇×C₇₁ ⋊ C₄
Sylow	C₇₁ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₇₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₃₃₇
commutator, hypocenter	C₄₇×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₇×C₇₁	C₂
LCS	G ▷ C₄₇×C₇₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 13348

A minimal faithful permutation representation — 2 generators on 122 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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122);(1,3,2,4)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(53,122)(54,121)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)

Polycyclic presentation:

2,47,71,2|a^2=d,b^a=b^46,c^a=c^70

Conjugacy classes & character table

3340 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₆₇₄
maximal normal subgroup of	Dic₆₆₇₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13348, 4);

Dic3337 #27211150