C₅×C₃⋊(C₃⋊#64408) #27190302

order	13140
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊(C₃⋊#64408)
unambiguous semidirect product	C₃⋊(C₅×C₃⋊#64408)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₇₃ ⋊ C₄
derived, Hall	C₃²×C₇₃ ⋊ C₄×C₅
Hall	C₅×C₇₃ ⋊ C₃⋊Dic₃
Sylow	C₇₃ ⋊ C₅×C₃⋊Dic₃
Hall	C₃²×C₅ ⋊ #64408
Sylow	C₃² ⋊ C₅×#64408

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(C₃⋊#64408)
commutator, hypocenter	C₃²×C₇₃	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇₃	C₄
prime radical	C₃²×C₅×D₇₃	C₂
cosocle	C₃²×D₇₃	C₂×C₅
LCS	G ▷ C₃²×C₇₃
UCS	C₁ ◁ C₅

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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 136);

C5×C3⋊(C3⋊#64408) #27190302