C₃²×#16814878 #45686401

order	33300
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×#16814878

Splittings

Fitting, socle, Hall	C₃²×C₅²×C₃₇ ⋊ C₄
derived, Hall	C₅²×C₃₇ ⋊ C₃²×C₄
Hall	C₃²×C₃₇ ⋊ #1043
Hall	C₃²×C₅² ⋊ #3866
Sylow	C₃₇ ⋊ C₃²×#1043
Sylow	C₅² ⋊ C₃²×#3866

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	#16814878
commutator, hypocenter	C₅²×C₃₇	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₃²×C₅⋊D₁₈₅	C₂
cosocle	C₅⋊D₁₈₅	C₂×C₃²
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₃²

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 8325

A minimal faithful permutation representation — 3 generators on 53 points:

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

Conjugacy classes & character table

2115 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 674);

C32×#16814878 #45686401