C₃×D₃₇⋊((C₅⋊D₅)⋊C₃) #45686203

order	33300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃×C₅

Aliases

direct product

C₃×D₃₇⋊((C₅⋊D₅)⋊C₃)

Splittings

Fitting, socle	C₃×C₅²×C₃₇ ⋊ C₂²×C₃
derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₃²
Hall	C₃×C₅²⋊(C₃₇⋊C₃) ⋊ C₂²
Hall	C₅⋊D₅×D₃₇ ⋊ C₃²
Sylow	C₃₇ ⋊ C₂×C₃×(C₅⋊D₅)⋊C₃
Sylow	C₅² ⋊ C₂×C₃×D₃₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₃₇⋊((C₅⋊D₅)⋊C₃)
commutator, cosocle, hypocenter	C₅²×C₃₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₃₇	C₂²×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₃

160 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1665

A minimal faithful permutation representation — 2 generators on 55 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 476);