C₂×C₅×D₁₀₀₃ #34095421

order	20060
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₁₀₀₃
unambiguous semidirect product	(C₂×C₁₇)⋊(C₅×D₅₉)
unambiguous semidirect product	(C₂×C₅₉)⋊(C₅×D₁₇)

Splittings

Fitting, socle	C₂×C₅×C₁₇×C₅₉ ⋊ C₂
derived, Hall	C₁₇×C₅₉ ⋊ C₂²×C₅
Hall	C₅×C₁₇×C₅₉ ⋊ C₂²
Hall	C₅×C₅₉ ⋊ C₂×D₁₇
Hall	C₅×C₁₇ ⋊ C₂×D₅₉
Sylow	C₅₉ ⋊ C₂×C₅×D₁₇
Sylow	C₁₇ ⋊ C₂×C₅×D₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₁₀₀₃
commutator, cosocle, hypocenter	C₁₇×C₅₉	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₇×C₅₉	C₂
LCS	G ▷ C₁₇×C₅₉
UCS	C₁ ◁ C₂×C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 10030

A minimal faithful permutation representation — 2 generators on 83 points:

(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)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78,79,80,81);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,61,63,65,67,69,71,73,75,76,74,72,70,68,66,64,62)(82,83)

Conjugacy classes & character table

5030 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20060, 31);

C2×C5×D1003 #34095421