C₃×Dic₁₅×C₇₃ #27190237

order	13140
isoclinism	isoclinic class of D₁₅
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₇₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×Dic₁₅×C₇₃
unambiguous semidirect product	C₃⋊(C₃×Dic₅×C₇₃)

Splittings

derived	C₃×C₅ ⋊ C₃×C₄×C₇₃
Hall	C₃²×C₅×C₇₃ ⋊ C₄
Hall	C₃²×C₇₃ ⋊ Dic₅
Hall	C₅×C₇₃ ⋊ C₃×Dic₃
Hall	C₃²×C₅ ⋊ C₄×C₇₃
Sylow	C₃² ⋊ Dic₅×C₇₃
Sylow	C₅ ⋊ C₃×Dic₃×C₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇₃	D₁₅
commutator, hypocenter	C₃×C₅	C₃×C₄×C₇₃
Frattini	C₂	C₃×D₁₅×C₇₃
Fitting, socle, prime radical	C₂×C₃²×C₅×C₇₃	C₂
cosocle	C₂×C₃×C₅	C₂×C₃×C₇₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₇₃

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 4380

A minimal faithful permutation representation — 2 generators on 88 points:

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

Conjugacy classes & character table

3942 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 71);

C3×Dic15×C73 #27190237