C₅×C₃⋊Dic₃×C₇₃ #27190239

order	13140
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₇₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊Dic₃×C₇₃
unambiguous semidirect product	C₃⋊(C₅×Dic₃×C₇₃)

Splittings

derived, Sylow	C₃² ⋊ C₄×C₅×C₇₃
Hall	C₃²×C₅×C₇₃ ⋊ C₄
Hall	C₃²×C₇₃ ⋊ C₄×C₅
Hall	C₃²×C₅ ⋊ C₄×C₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅×C₇₃	C₃⋊D₃
commutator, hypocenter	C₃²	C₄×C₅×C₇₃
Frattini	C₂	C₅×C₃⋊D₃×C₇₃
Fitting, socle, prime radical	C₂×C₃²×C₅×C₇₃	C₂
cosocle	C₂×C₃²	C₂×C₅×C₇₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅×C₇₃

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 13140

A minimal faithful permutation representation — 3 generators on 88 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77)(79,80)(82,83)(84,85,86,87,88);(81,82,83);(78,79,80)

Conjugacy classes & character table

4380 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 73);

C5×C3⋊Dic3×C73 #27190239