#27211840

order	13420
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived, Hall	C₁₁×C₆₁ ⋊ C₄×C₅
Hall	#16809821 ⋊ C₄
Hall	Dic₆₇₁ ⋊ C₅
Sylow	C₆₁ ⋊ Dic₁₁⋊C₅
Sylow	C₁₁ ⋊ Dic₆₁⋊C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#20269529
commutator, hypocenter	C₁₁×C₆₁	C₄×C₅
Fitting, socle, cosocle	C₂×C₁₁×C₆₁	C₂×C₅
prime radical	C₂×#16809821	C₂
LCS	G ▷ C₁₁×C₆₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 2684

A minimal faithful permutation representation — 2 generators on 76 points:

(2,6,7,20,25,3,4,11,13,35)(5,17,18,26,46,9,10,31,32,43)(8,27,28,29,42,14,15,45,24,47)(12,40,41,60,19,22,23,54,55,34)(16,37,38,59,49,30,21,50,51,36)(33,53,52,56,57,39,48,61,44,58)(63,67,68,66,71,64,65,72,69,70)(73,74,75,76);(1,2,5,11,21,39,31,10,19,25,42,55,59,48,38,52,57,28,46,49,13,26,44,54,40,30,14,6,12,24,27,15,29,22,4,8,16,23,41,56,32,20,36,43,45,58,61,50,53,51,60,47,35,34,17,18,33,37,7,9,3)(62,63,66,71,65,68,72,67,70,69,64)

Polycyclic presentation:

2,5,11,61,2|a^2=e,c^a=c^10,d^a=d^60,c^b=c^4,d^b=d^34

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 32);