C₂×(C₁₁⋊C₅)⋊D₆₁ #27211870

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

Aliases

direct product	C₂×(C₁₁⋊C₅)⋊D₆₁
unambiguous semidirect product	(C₂×C₁₁⋊C₅)⋊D₆₁

Splittings

Fitting, socle	C₂×C₁₁×C₆₁ ⋊ C₂×C₅
derived, Hall	C₁₁×C₆₁ ⋊ C₂²×C₅
Hall	C₁₁⋊C₅×C₆₁ ⋊ C₂²
Hall	C₂×D₆₇₁ ⋊ C₅
Sylow	C₆₁ ⋊ C₂×AGL(1,𝔽₁₁)
Hall	C₁₁⋊C₅ ⋊ C₂×D₆₁
Sylow	C₁₁ ⋊ C₂×C₅×D₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	(C₁₁⋊C₅)⋊D₆₁
commutator, cosocle, hypocenter	C₁₁×C₆₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₁₁×C₆₁	C₂×C₅
LCS	G ▷ C₁₁×C₆₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 1342

A minimal faithful permutation representation — 2 generators on 74 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 62);

C2×(C11⋊C5)⋊D61 #27211870