C₂×C₅×C₇⋊D₂₅₉ #49136458

order	36260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₇

Aliases

direct product	C₂×C₅×C₇⋊D₂₅₉
unambiguous semidirect product	(C₂×C₇)⋊(C₅×D₂₅₉)
unambiguous semidirect product	(C₂×C₃₇)⋊(C₅×C₇⋊D₇)

Splittings

Fitting, socle	C₂×C₅×C₇²×C₃₇ ⋊ C₂
derived, Hall	C₇²×C₃₇ ⋊ C₂²×C₅
Hall	C₅×C₇²×C₃₇ ⋊ C₂²
Hall	C₅×C₇² ⋊ C₂×D₃₇
Hall	C₅×C₃₇ ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×C₅×D₃₇
Sylow	C₃₇ ⋊ C₂×C₅×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₇⋊D₂₅₉
commutator, cosocle, hypocenter	C₇²×C₃₇	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇²×C₃₇	C₂
LCS	G ▷ C₇²×C₃₇
UCS	C₁ ◁ C₂×C₅

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 18130

A minimal faithful permutation representation — 3 generators on 58 points:

(3,4,5,6,7)(9,10)(11,12)(13,14)(16,17)(18,19)(20,21)(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);(1,2)(15,16,18,20,21,19,17)(22,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24);(8,9,11,13,14,12,10)

Conjugacy classes & character table

9080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36260, 104);

C2×C5×C7⋊D259 #49136458