(C₅⋊D₅)⋊C₃×D₃₇⋊C₃ #45686147

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

Aliases

direct product	(C₅⋊D₅)⋊C₃×D₃₇⋊C₃
unambiguous semidirect product	(C₅⋊D₅×D₃₇⋊C₃)⋊C₃
unambiguous semidirect product	(D₃₇⋊((C₅⋊D₅)⋊C₃))⋊C₃
unambiguous semidirect product	(D₃₇×(C₅⋊D₅)⋊C₃)⋊C₃

Splittings

Fitting, socle, derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₃²
Hall	C₅²⋊C₃×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, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²×C₃₇	C₂²×C₃²
LCS	G ▷ C₅²×C₃₇
UCS	C₁

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 52
transitive degree	μ_t(G) = 555

A minimal faithful permutation representation — 2 generators on 52 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 420);