C₉₇×#2102 #27138060

order	12416
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉₇×#2102

Subgroups

	subgroup	G/H
center	C₄×C₉₇	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
socle	C₂×C₉₇	#400
prime radical	C₂³×C₉₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ G

158 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 113
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 3 generators on 113 points:

(1,2,5,4,6,8,10,14)(3,9,12,16,13,11,15,7)(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);(1,3,10,12,6,13,5,15)(2,7,14,9,8,16,4,11);(1,4,5,8,6,14,10,2)(3,11,12,7,13,9,15,16)

Conjugacy classes & character table

2522 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 834);