C₂×D₃×C₂₃×C₄₇ #27188451

order	12972
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₂₃×C₄₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×C₂₃×C₄₇
unambiguous semidirect product	C₃⋊(C₂²×C₂₃×C₄₇)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₂₃×C₄₇)

Splittings

Fitting, socle	C₂×C₃×C₂₃×C₄₇ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂²×C₂₃×C₄₇
Hall	C₃×C₂₃×C₄₇ ⋊ C₂²
Hall	C₃×C₄₇ ⋊ C₂²×C₂₃
Hall	C₃×C₂₃ ⋊ C₂²×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₃×C₄₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₂₃×C₄₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₂₃×C₄₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₂₃×C₄₇

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6486

A minimal faithful permutation representation — 2 generators on 75 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);(1,2,3)(74,75)

Conjugacy classes & character table

6486 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12972, 31);

C2×D3×C23×C47 #27188451