C₅⋊D₅×#66526⋊C₂ #45686177

order	33300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₅⋊D₅×#66526⋊C₂
unambiguous semidirect product	(C₅×#66526⋊C₂)⋊D₅

Splittings

Fitting, socle, derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₉
Hall	C₅²×#66526 ⋊ C₂²
Hall	C₅⋊D₅×D₃₇ ⋊ C₉
Hall	#66526 ⋊ C₂×C₅⋊D₅
Sylow	C₃₇ ⋊ C₂×C₉×C₅⋊D₅
Sylow	C₅² ⋊ C₂×#66526⋊C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₃₇	C₂²×C₉
prime radical	C₅⋊D₅×D₃₇⋊C₃	C₃
cosocle	C₅²×C₃₇⋊C₃	C₂²×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 925

A minimal faithful permutation representation — 3 generators on 47 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 450);

C5⋊D5×#66526⋊C2 #45686177