Dic₂₈₉₁ #26040581

order	11564
automorphism	Aut(G) order 14084952
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₈₉₁
unambiguous semidirect product	C₄₉⋊Dic₅₉
unambiguous semidirect product	C₅₉⋊Dic₄₉

Splittings

derived, Hall	C₄₉×C₅₉ ⋊ C₄
Sylow	C₅₉ ⋊ Dic₄₉
Sylow	C₄₉ ⋊ Dic₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₈₉₁
commutator, hypocenter	C₄₉×C₅₉	C₄
Frattini	C₂×C₇	D₄₁₃
Fitting, cosocle	C₂×C₄₉×C₅₉	C₂
socle, prime radical	C₂×C₇×C₅₉	D₇
LCS	G ▷ C₄₉×C₅₉
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 11564

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

Polycyclic presentation:

2,7,59,2,7|a^2=d,b^7=e^6,b^a=b^6e,c^a=c^58,e^a=e^6

Conjugacy classes & character table

2894 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₈₉₁, Dic₈₆₇₃, C₅₉⋊((C₄₉⋊C₃)⋊C₄), …
maximal normal subgroup of	C₃×Dic₂₈₉₁, C₅₉⋊((C₄₉⋊C₃)⋊C₄), Dic₅₇₈₂
maximal quotient of	C₃×Dic₂₈₉₁, Dic₈₆₇₃, Dic₁₄₄₅₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11564, 4);

Dic2891 #26040581