Dic₇₇₉₁ #44528317

order	31164
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₇₉₁
unambiguous semidirect product	C₃⋊Dic₂₅₉₇
unambiguous semidirect product	C₄₉⋊Dic₁₅₉

Splittings

derived, Hall	C₃×C₄₉×C₅₃ ⋊ C₄
Hall	C₄₉×C₅₃ ⋊ Dic₃
Hall	C₃×C₅₃ ⋊ Dic₄₉
Hall	C₃×C₄₉ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₄₇
Sylow	C₄₉ ⋊ Dic₁₅₉
Sylow	C₃ ⋊ Dic₂₅₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₇₉₁
commutator, hypocenter	C₃×C₄₉×C₅₃	C₄
Frattini	C₂×C₇	D₁₁₁₃
Fitting, cosocle	C₂×C₃×C₄₉×C₅₃	C₂
socle, prime radical	C₂×C₃×C₇×C₅₃	D₇
LCS	G ▷ C₃×C₄₉×C₅₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 31164

A minimal faithful permutation representation — 2 generators on 109 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);(1,3,2,4)(6,7)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(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:

2,3,7,53,2,7|a^2=e,c^7=f^6,b^a=b^2,c^a=c^6f,d^a=d^52,f^a=f^6

Conjugacy classes & character table

7794 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₅₈₂
maximal normal subgroup of	Dic₁₅₅₈₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31164, 9);

Dic7791 #44528317