3₋¹⁺²×C₁₃₉ #16836722

order	3753
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₁₃₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	3₋¹⁺²×C₁₃₉
unambiguous semidirect product	C₉⋊(C₃×C₁₃₉)

Subgroups

	subgroup	G/H
center, socle	C₃×C₁₃₉	C₃²
commutator, Frattini, cosocle	C₃	C₃²×C₁₃₉
prime radical	C₃²×C₁₃₉	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₃₉ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 148
transitive degree	μ_t(G) = 1251

A minimal faithful permutation representation — 2 generators on 148 points:

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

Conjugacy classes & character table

1529 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	AΣL(1,𝔽₂₇)×C₁₃₉, #731×C₁₃₉, #733×C₁₃₉, …
maximal normal subgroup of	AΣL(1,𝔽₂₇)×C₁₃₉, #731×C₁₃₉, #733×C₁₃₉, …
maximal quotient of	C₃²⋊C₉×C₁₃₉, C₉⋊C₉×C₁₃₉, C₃×3₋¹⁺²×C₁₃₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3753, 4);

3−1+2×C139 #16836722