C₃₇×#10624905 #41004694

order	26973
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₃₇ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₃₇×#10624905

Subgroups

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃₇×C₃²⋊H(3,𝔽₃)	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

788 conjugacy classes of subgroups; only the 446 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 446 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 2997

A minimal faithful permutation representation — 4 generators on 91 points:

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

Conjugacy classes & character table

4477 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 465);