C₂×C₅×C₁₁²⋊C₃ #16813832

order	3630
isoclinism	isoclinic class of C₁₁²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁₁

Aliases

direct product	C₂×C₅×C₁₁²⋊C₃
unambiguous semidirect product	(C₂×C₅×C₁₁²)⋊C₃

Splittings

Fitting, socle, Hall	C₂×C₅×C₁₁² ⋊ C₃
derived, Sylow	C₁₁² ⋊ C₂×C₃×C₅
Hall	C₅×C₁₁² ⋊ C₂×C₃
Hall	C₂×C₁₁² ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₁₁²⋊C₃
commutator, cosocle, hypocenter	C₁₁²	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₁²	C₃
LCS	G ▷ C₁₁²
UCS	C₁ ◁ C₂×C₅

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 330

A minimal faithful permutation representation — 2 generators on 40 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×D₃×C₁₁²⋊C₃, C₂×C₃×C₅×C₁₁²⋊C₃, C₂×(C₁₁²⋊C₃)⋊C₂₅, …
maximal normal subgroup of	C₂×C₃×C₅×C₁₁²⋊C₃, C₂×(C₁₁²⋊C₃)⋊C₂₅, C₂×C₂₅×C₁₁²⋊C₃, …
maximal quotient of	C₂×C₅×C₁₁²⋊C₉, C₅×D₃×C₁₁²⋊C₃, C₂×C₃×C₅×C₁₁²⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3630, 52);

C2×C5×C112⋊C3 #16813832