#37553534

order	23478
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₄₃ ⋊ C₂×C₃
Hall	C₇×#13131975 ⋊ C₂
Hall	D₃₉₁₃ ⋊ C₃
Hall	#13131975 ⋊ D₇
Hall	C₁₃×C₄₃ ⋊ C₃×D₇
Hall	C₇×C₄₃ ⋊ D₁₃⋊C₃
Hall	C₇×C₁₃ ⋊ D₄₃⋊C₃
Sylow	C₄₃ ⋊ (C₁₃⋊C₃)⋊D₇
Sylow	C₁₃ ⋊ (C₄₃⋊C₃)⋊D₇
Sylow	C₇ ⋊ #16809818

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₄₃	C₂×C₃
LCS	G ▷ C₇×C₁₃×C₄₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 3913

A minimal faithful permutation representation — 2 generators on 63 points:

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

Polycyclic presentation:

2,3,7,13,43|c^a=c^6,d^a=d^12,e^a=e^42,d^b=d^3,e^b=e^6

Conjugacy classes & character table

666 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23478, 123);