C₂×#34038670 #49180677

order	37820
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#34038670

Splittings

Fitting, socle	C₂×C₃₁×C₆₁ ⋊ C₂×C₅
derived, Hall	C₃₁×C₆₁ ⋊ C₂²×C₅
Hall	#22628160 ⋊ C₂²
Hall	C₂×D₁₈₉₁ ⋊ C₅
Sylow	C₆₁ ⋊ C₂×D₃₁⋊C₅
Sylow	C₃₁ ⋊ C₂×D₆₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#34038670
commutator, cosocle, hypocenter	C₃₁×C₆₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃₁×C₆₁	C₂×C₅
LCS	G ▷ C₃₁×C₆₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 94
transitive degree	μ_t(G) = 3782

A minimal faithful permutation representation — 2 generators on 94 points:

(4,8,9,20,7,5,6,13,14,11)(10,25,26,27,12,16,17,29,31,19)(15,32,24,30,18,22,23,33,21,28)(35,39,40,54,52,36,37,45,47,44)(38,41,51,58,63,42,43,49,67,77)(46,74,75,84,69,56,57,62,91,53)(48,81,82,79,80,60,61,83,59,78)(50,76,87,94,86,64,65,93,92,70)(55,90,85,71,88,72,73,89,68,66);(1,2)(3,4,7,12,14,21,31,25,9,15,24,32,26,28,16,8,6,10,18,29,23,33,22,13,17,27,30,20,19,11,5)(34,35,38,44,53,70,77,51,68,47,59,73,91,65,74,92,87,80,67,56,37,41,50,66,89,75,60,45,55,61,83,82,81,72,40,48,62,85,88,64,43,39,46,58,78,93,94,57,76,84,90,79,54,71,49,63,86,69,52,42,36)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37820, 57);

C2×#34038670 #49180677