Dic₈₇₁₅ #46852829

order	34860
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₇₁₅
unambiguous semidirect product	C₃⋊Dic₂₉₀₅
unambiguous semidirect product	C₇⋊Dic₁₂₄₅
unambiguous semidirect product	C₈₃⋊Dic₁₀₅

Splittings

derived, Hall	C₃×C₅×C₇×C₈₃ ⋊ C₄
Hall	C₅×C₇×C₈₃ ⋊ Dic₃
Hall	C₃×C₇×C₈₃ ⋊ Dic₅
Hall	C₃×C₅×C₈₃ ⋊ Dic₇
Hall	C₇×C₈₃ ⋊ Dic₁₅
Hall	C₅×C₈₃ ⋊ Dic₂₁
Hall	C₃×C₈₃ ⋊ Dic₃₅
Hall	C₃×C₅×C₇ ⋊ Dic₈₃
Sylow	C₈₃ ⋊ Dic₁₀₅
Hall	C₅×C₇ ⋊ Dic₂₄₉
Hall	C₃×C₇ ⋊ Dic₄₁₅
Hall	C₃×C₅ ⋊ Dic₅₈₁
Sylow	C₇ ⋊ Dic₁₂₄₅
Sylow	C₅ ⋊ Dic₁₇₄₃
Sylow	C₃ ⋊ Dic₂₉₀₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₇₁₅
commutator, hypocenter	C₃×C₅×C₇×C₈₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₇×C₈₃	C₂
LCS	G ▷ C₃×C₅×C₇×C₈₃
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 34860

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

Polycyclic presentation:

2,3,5,7,83,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^82

Conjugacy classes & character table

8718 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34860, 17);

Dic8715 #46852829