Dic₇₄₃₇ #44446553

order	29748
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₄₃₇
unambiguous semidirect product	C₃⋊Dic₂₄₇₉
unambiguous semidirect product	C₆₇⋊Dic₁₁₁

Splittings

derived, Hall	C₃×C₃₇×C₆₇ ⋊ C₄
Hall	C₃₇×C₆₇ ⋊ Dic₃
Hall	C₃×C₆₇ ⋊ Dic₃₇
Hall	C₃×C₃₇ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₁₁₁
Sylow	C₃₇ ⋊ Dic₂₀₁
Sylow	C₃ ⋊ Dic₂₄₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₄₃₇
commutator, hypocenter	C₃×C₃₇×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₃₇×C₆₇	C₂
LCS	G ▷ C₃×C₃₇×C₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 29748

A minimal faithful permutation representation — 2 generators on 111 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);(1,3,2,4)(6,7)(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)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2,3,37,67,2|a^2=e,b^a=b^2,c^a=c^36,d^a=d^66

Conjugacy classes & character table

7440 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₈₇₄
maximal normal subgroup of	Dic₁₄₈₇₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29748, 12);

Dic7437 #44446553