C₂×C₃⋊#14524208 #27188917

order	12996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃⋊#14524208
unambiguous semidirect product	(C₂×C₃)⋊#14524208

Splittings

Fitting, socle	C₂×C₃×C₁₉² ⋊ C₂×C₃
derived	C₃×C₁₉² ⋊ C₂²×C₃
Hall	C₃×#11762245 ⋊ C₂²
Sylow	C₁₉² ⋊ C₂×C₃×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#14524208
commutator, cosocle, hypocenter	C₃×C₁₉²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₉²	C₂×C₃
LCS	G ▷ C₃×C₁₉²
UCS	C₁ ◁ C₂

196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 2 generators on 43 points:

(2,3)(5,9,10,6,7,11)(8,17,14,12,13,19)(15,20,22,21,18,16)(24,25,27)(26,29,30)(28,32,31)(33,37,35)(34,38,36)(39,41,40);(1,2,3)(4,5,8,14,20,17,21,7,11,16,22,10,9,15,13,18,19,12,6)(23,24,26,30,35,40,37,25,28,33,38,27,31,36,29,34,39,41,32)(42,43)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 260);

C2×C3⋊#14524208 #27188917