C₃⋊(C₃⋊#13091734) #27190307

order	13140
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product

C₃⋊(C₃⋊#13091734)

Splittings

Fitting, socle, derived, Hall	C₃²×C₅×C₇₃ ⋊ C₄
Hall	C₃²×C₇₃ ⋊ AGL(1,𝔽₅)
Hall	C₅×C₇₃ ⋊ C₃⋊Dic₃
Sylow	C₇₃ ⋊ C₃⋊ΓL(1,𝔽₁₆)
Hall	C₃²×C₅ ⋊ Dic₇₃
Sylow	C₃² ⋊ #13091734
Sylow	C₅ ⋊ C₃⋊Dic₂₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅×C₇₃	C₄
prime radical, cosocle	C₃²×D₅×C₇₃	C₂
LCS	G ▷ C₃²×C₅×C₇₃
UCS	C₁

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 3285

A minimal faithful permutation representation — 3 generators on 84 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(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)(64,65)(66,67)(68,69)(70,71)(72,73)(75,76,77,78)(80,81)(83,84);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(74,75,76,78,77)(82,83,84);(79,80,81)

Polycyclic presentation:

2^2,3^2,5,73|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^72,e^b=e^4

Conjugacy classes & character table

1317 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 141);

C3⋊(C3⋊#13091734) #27190307