C₂×D₉×C₁₉² #27188757

order	12996
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₉² · Schur multiplier (H₂) = C₂×C₁₉

Aliases

direct product	C₂×D₉×C₁₉²
unambiguous semidirect product	C₉⋊(C₂²×C₁₉²)
unambiguous semidirect product	(C₂×C₉)⋊(C₂×C₁₉²)

Splittings

Fitting	C₂×C₉×C₁₉² ⋊ C₂
derived, Sylow	C₉ ⋊ C₂²×C₁₉²
Hall	C₉×C₁₉² ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉²	D₉
commutator, cosocle, hypocenter	C₉	C₂²×C₁₉²
Frattini	C₃	C₂×D₃×C₁₉²
Fitting	C₂×C₉×C₁₉²	C₂
socle	C₂×C₃×C₁₉²	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₁₉²

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 6498

A minimal faithful permutation representation — 2 generators on 49 points:

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

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, 100);

C2×D9×C192 #27188757