Dic₃₃₉₇ #28331120

order	13588
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₃₉₇
unambiguous semidirect product	C₄₃⋊Dic₇₉
unambiguous semidirect product	C₇₉⋊Dic₄₃

Splittings

derived, Hall	C₄₃×C₇₉ ⋊ C₄
Sylow	C₇₉ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₃₉₇
commutator, hypocenter	C₄₃×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₃×C₇₉	C₂
LCS	G ▷ C₄₃×C₇₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 13588

A minimal faithful permutation representation — 2 generators on 126 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)(48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126);(1,3,2,4)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(49,126)(50,125)(51,124)(52,123)(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:

2,43,79,2|a^2=d,b^a=b^42,c^a=c^78

Conjugacy classes & character table

3400 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₇₉₄
maximal normal subgroup of	Dic₆₇₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13588, 4);

Dic3397 #28331120