C₂×#30659246 #45685993

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

Aliases

direct product

C₂×#30659246

Splittings

derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₉
Hall	C₃₇×C₅²⋊C₉ ⋊ C₂²
Hall	C₂×C₅⋊D₁₈₅ ⋊ C₉
Hall	C₅²⋊C₉ ⋊ C₂×D₃₇
Sylow	C₃₇ ⋊ C₂×(C₅⋊D₅)⋊C₉
Sylow	C₅² ⋊ C₂×C₉×D₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#19149853
commutator, hypocenter	C₅²×C₃₇	C₂²×C₉
Frattini	C₃	C₂×#19149853
Fitting, socle	C₂×C₃×C₅²×C₃₇	C₂×C₃
prime radical	C₂×C₃×C₅⋊D₁₈₅	C₃
cosocle	C₃×C₅²×C₃₇	C₂²×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₂×C₃

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 3330

A minimal faithful permutation representation — 2 generators on 63 points:

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

Conjugacy classes & character table

1248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 266);

C2×#30659246 #45685993