C₂²×#16809823 #27211880

order	13420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂²×#16809823

Splittings

Fitting, socle, Hall	C₂²×C₁₁×C₆₁ ⋊ C₅
derived, Hall	C₁₁×C₆₁ ⋊ C₂²×C₅
Hall	C₂²×C₆₁ ⋊ C₁₁⋊C₅
Sylow	C₆₁ ⋊ C₂²×C₁₁⋊C₅
Hall	C₂²×C₁₁ ⋊ C₆₁⋊C₅
Sylow	C₁₁ ⋊ C₂²×C₆₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#16809823
commutator, cosocle, hypocenter	C₁₁×C₆₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₁₁×C₆₁	C₅
LCS	G ▷ C₁₁×C₆₁
UCS	C₁ ◁ C₂²

40 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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 72);

C22×#16809823 #27211880