Dic₈₂₉₅ #45684389

order	33180
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₂₉₅
unambiguous semidirect product	C₃⋊Dic₂₇₆₅
unambiguous semidirect product	C₇⋊Dic₁₁₈₅
unambiguous semidirect product	C₇₉⋊Dic₁₀₅

Splittings

derived, Hall	C₃×C₅×C₇×C₇₉ ⋊ C₄
Hall	C₅×C₇×C₇₉ ⋊ Dic₃
Hall	C₃×C₇×C₇₉ ⋊ Dic₅
Hall	C₃×C₅×C₇₉ ⋊ Dic₇
Hall	C₇×C₇₉ ⋊ Dic₁₅
Hall	C₅×C₇₉ ⋊ Dic₂₁
Hall	C₃×C₇₉ ⋊ Dic₃₅
Hall	C₃×C₅×C₇ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₀₅
Hall	C₅×C₇ ⋊ Dic₂₃₇
Hall	C₃×C₇ ⋊ Dic₃₉₅
Hall	C₃×C₅ ⋊ Dic₅₅₃
Sylow	C₇ ⋊ Dic₁₁₈₅
Sylow	C₅ ⋊ Dic₁₆₅₉
Sylow	C₃ ⋊ Dic₂₇₆₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₂₉₅
commutator, hypocenter	C₃×C₅×C₇×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₇×C₇₉	C₂
LCS	G ▷ C₃×C₅×C₇×C₇₉
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 33180

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

Polycyclic presentation:

2,3,5,7,79,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^78

Conjugacy classes & character table

8298 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33180, 20);

Dic8295 #45684389