C₁₉²×C₂²⋊C₉ #27188751

order	12996
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₉² · Schur multiplier (H₂) = C₂×C₁₉

Aliases

direct product	C₁₉²×C₂²⋊C₉
unambiguous semidirect product	C₂²⋊(C₉×C₁₉²)

Splittings

derived, Sylow	C₂² ⋊ C₉×C₁₉²
Hall	C₂²×C₁₉² ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₉²	A₄
commutator, hypocenter	C₂²	C₉×C₁₉²
Frattini	C₃	A₄×C₁₉²
Fitting, socle, prime radical	C₂²×C₃×C₁₉²	C₃
cosocle	C₂²×C₃	C₃×C₁₉²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₁₉²

176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 6498

A minimal faithful permutation representation — 2 generators on 51 points:

(2,3,4)(5,6,8,7,9,11,10,12,13)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(1,4)(2,3)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)

Conjugacy classes & character table

4332 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 94);

C192×C22⋊C9 #27188751