D₅×C₁₁² #11925907

order	1210
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁² · Schur multiplier (H₂) = C₁₁

Aliases

direct product	D₅×C₁₁²
unambiguous semidirect product	C₅⋊(C₂×C₁₁²)

Fitting, socle, Hall	C₅×C₁₁² ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₁²

	subgroup	G/H
center, hypercenter	C₁₁²	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₁²
Frattini	C₁	G
Fitting, socle	C₅×C₁₁²	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₁²

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 605

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16);(2,5)(3,4)(17,18,19,20,21,22,23,24,25,26,27)

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1210, 24);