#37613200

order	24738
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₁₉×C₃₁ ⋊ C₂×C₃
Hall	#27136504 ⋊ C₂
Hall	D₄₁₂₃ ⋊ C₃
Hall	C₁₉×C₃₁ ⋊ AGL(1,𝔽₇)
Hall	C₇×C₃₁ ⋊ D₁₉⋊C₃
Hall	C₇×C₁₉ ⋊ D₃₁⋊C₃
Sylow	C₃₁ ⋊ #11716328
Sylow	C₁₉ ⋊ #13049960
Sylow	C₇ ⋊ #16812975

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₉×C₃₁	C₂×C₃
LCS	G ▷ C₇×C₁₉×C₃₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 4123

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

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

Polycyclic presentation:

2,3,7,19,31|c^a=c^6,d^a=d^18,e^a=e^30,c^b=c^2,d^b=d^11,e^b=e^5

Conjugacy classes & character table

693 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24738, 120);