#44545196

order	32046
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₄₉×C₁₀₉ ⋊ C₂×C₃
Hall	#30630916 ⋊ C₂
Hall	D₅₃₄₁ ⋊ C₃
Sylow	C₁₀₉ ⋊ D₄₉⋊C₃
Sylow	C₄₉ ⋊ D₁₀₉⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₁₀₉	C₂×C₃
Frattini	C₇	#17987086
socle	C₇×C₁₀₉	AGL(1,𝔽₇)
LCS	G ▷ C₄₉×C₁₀₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 158
transitive degree	μ_t(G) = 5341

A minimal faithful permutation representation — 2 generators on 158 points:

(2,10,12,4,5,13)(3,22,24,8,9,25)(6,34,21,15,16,43)(7,39,40,19,20,49)(11,18,17,23,28,36)(14,37,46,32,33,47)(26,38,41,44,27,30)(29,42,31,45,35,48)(51,55,56,52,53,57)(54,63,64,58,59,73)(60,76,77,66,67,90)(61,75,71,69,65,62)(68,95,96,79,80,113)(70,93,94,82,78,87)(72,74,92,84,85,88)(81,118,119,98,99,140)(83,116,117,101,97,110)(86,91,115,103,104,111)(89,129,127,107,108,106)(100,145,146,121,122,158)(102,143,144,123,120,137)(105,114,142,124,125,138)(109,150,151,130,131,133)(112,154,148,134,135,128)(126,141,152,147,132,157)(136,156,149,139,155,153);(1,23,20,31,30,35,4,8,45,14,13,40,15,19,22,46,5,34,36,28,43,24,44,38,37,33,27,26,25,21,18,17,16,10,47,9,7,6,49,12,32,29,3,2,42,41,48,39,11)(50,51,54,60,68,81,100,119,142,154,150,143,118,141,130,107,85,59,65,73,87,106,128,149,151,117,77,92,93,76,91,112,136,125,80,97,108,111,113,137,147,124,103,84,71,56,55,61,70,83,102,122,146,152,155,156,157,158,145,123,101,82,69,53,57,62,72,86,105,126,144,96,115,129,116,95,114,139,134,104,67,78,88,90,110,133,153,148,127,94,64,75,63,74,89,109,132,99,120,131,135,138,140,121,98,79,66,58,52)

Polycyclic presentation:

2,3,7,109,7|c^7=e^6,c^a=c^6e,d^a=d^108,e^a=e^6,c^b=c^2e^3,d^b=d^63,e^b=e^2

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32046, 23);