C₁₉×C₂²⋊#4294 #27188752

order	12996
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₉×C₂²⋊#4294
unambiguous semidirect product	C₂²⋊(C₁₉×#4294)

Splittings

derived	C₂²×C₁₉ ⋊ C₉×C₁₉
Hall	C₂²×C₁₉² ⋊ C₉
Sylow	C₁₉² ⋊ C₂²⋊C₉
Sylow	C₂² ⋊ C₁₉×#4294

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₉	C₂²⋊(C₁₉⋊C₃)
commutator, hypocenter	C₂²×C₁₉	C₉×C₁₉
Frattini	C₃	C₁₉×C₂²⋊(C₁₉⋊C₃)
Fitting, socle, prime radical	C₂²×C₃×C₁₉²	C₃
cosocle	C₂²×C₃×C₁₉	C₃×C₁₉
LCS	G ▷ C₂²×C₁₉
UCS	C₁ ◁ C₃×C₁₉

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 342

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

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

Conjugacy classes & character table

1596 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 95);

C19×C22⋊#4294 #27188752