C₂×C₅×#66524 #16809665

order	3330
isoclinism	isoclinic class of C₃₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂×C₅×#66524

Splittings

derived, Sylow	C₃₇ ⋊ C₂×C₅×C₉
Hall	C₂×C₅×C₃₇ ⋊ C₉
Hall	C₅×C₃₇ ⋊ C₂×C₉
Hall	C₂×C₃₇ ⋊ C₅×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	C₃₇⋊C₃
commutator, hypocenter	C₃₇	C₂×C₅×C₉
Frattini	C₃	C₂×C₅×C₃₇⋊C₃
Fitting, socle, prime radical	C₂×C₃×C₅×C₃₇	C₃
cosocle	C₃×C₃₇	C₂×C₃×C₅
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×C₃×C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 3330

A minimal faithful permutation representation — 2 generators on 53 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₂₅×#66524, C₅×D₅×#66524, C₂×C₅²×#66524, …
maximal normal subgroup of	C₂×C₂₅×#66524, C₂×C₅²×#66524, C₂×C₅×C₇×#66524, …
maximal quotient of	C₂×C₂₅×#66524, C₅×D₅×#66524, C₂×C₅²×#66524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3330, 5);

C2×C5×#66524 #16809665