#34118564

order	20202
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₃₇ ⋊ C₂×C₃
Hall	#23758836 ⋊ C₂
Hall	C₁₃×D₂₅₉ ⋊ C₃
Hall	D₂₅₉ ⋊ C₁₃⋊C₃
Hall	C₁₃×C₃₇ ⋊ AGL(1,𝔽₇)
Hall	C₇×C₃₇ ⋊ C₂×C₁₃⋊C₃
Hall	C₇×C₁₃ ⋊ D₃₇⋊C₃
Sylow	C₃₇ ⋊ #10587721
Sylow	C₁₃ ⋊ #13095538
Sylow	C₇ ⋊ #15668359

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₃₇	C₂×C₃
LCS	G ▷ C₇×C₁₃×C₃₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 3367

A minimal faithful permutation representation — 2 generators on 57 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,19,17,18,29)(12,26,22,20,16,13)(21,37,34,31,28,35)(23,25,30,32,33,36)(39,40,42)(41,44,48)(43,46,45)(47,49,50)(52,55,56,53,54,57);(1,2,5,11,19,30,37,27,31,20,4,8,13,23,34,15,26,14,25,33,10,16,24,35,32,22,7,6,12,21,18,28,36,29,17,9,3)(38,39,41,42,45,44,48,50,49,40,43,47,46)(51,52,54,57,56,55,53)

Polycyclic presentation:

2,3,7,13,37|c^a=c^6,e^a=e^36,c^b=c^2,d^b=d^9,e^b=e^10

Conjugacy classes & character table

573 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20202, 94);