C₂×C₇×C₅⋊D₂₆₅ #49150754

order	37100
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₂×C₇×C₅⋊D₂₆₅
unambiguous semidirect product	(C₂×C₅)⋊(C₇×D₂₆₅)
unambiguous semidirect product	(C₂×C₅₃)⋊(C₇×C₅⋊D₅)

Splittings

Fitting, socle	C₂×C₅²×C₇×C₅₃ ⋊ C₂
derived, Hall	C₅²×C₅₃ ⋊ C₂²×C₇
Hall	C₅²×C₇×C₅₃ ⋊ C₂²
Hall	C₇×C₅₃ ⋊ C₂×C₅⋊D₅
Hall	C₅²×C₇ ⋊ C₂×D₅₃
Sylow	C₅₃ ⋊ C₂×C₇×C₅⋊D₅
Sylow	C₅² ⋊ C₂×C₇×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	C₅⋊D₂₆₅
commutator, cosocle, hypocenter	C₅²×C₅₃	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇×C₅₃	C₂
LCS	G ▷ C₅²×C₅₃
UCS	C₁ ◁ C₂×C₇

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 18550

A minimal faithful permutation representation — 3 generators on 72 points:

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

Conjugacy classes & character table

9296 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37100, 116);

C2×C7×C5⋊D265 #49150754