C₈₉×ΓL(1,𝔽₃₂) #28352280

order	13795
isoclinism	isoclinic class of ΓL(1,𝔽₃₂)
automorphism	?
homology	Abelianization (H₁) = C₅×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₈₉×ΓL(1,𝔽₃₂)
unambiguous semidirect product	C₃₁⋊(C₅×C₈₉)
unambiguous semidirect product	(C₃₁×C₈₉)⋊C₅

Splittings

Fitting, socle, Hall	C₃₁×C₈₉ ⋊ C₅
derived, Sylow	C₃₁ ⋊ C₅×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₈₉	ΓL(1,𝔽₃₂)
commutator, cosocle, hypocenter	C₃₁	C₅×C₈₉
Frattini	C₁	G
Fitting, socle	C₃₁×C₈₉	C₅
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₈₉

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 2759

A minimal faithful permutation representation — 2 generators on 120 points:

(2,3,5,8,4)(6,10,15,18,7)(9,13,14,12,11)(16,22,26,24,17)(19,23,21,25,20)(27,31,30,29,28)(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);(1,2,4,7,8,12,18,24,5,9,14,20,15,21,26,30,3,6,11,17,13,19,25,29,10,16,23,28,22,27,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₈₉×ΓL(1,𝔽₃₂), ΓL(1,𝔽₃₂)×D₈₉, C₈₉×D₃₁⋊C₅, …
maximal normal subgroup of	C₂×C₈₉×ΓL(1,𝔽₃₂), ΓL(1,𝔽₃₂)×D₈₉, C₈₉×D₃₁⋊C₅, …
maximal quotient of	C₂×C₈₉×ΓL(1,𝔽₃₂)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13795, 1);

C89×ΓL(1,𝔽32) #28352280