C₉₇×#1801 #27137759

order	12416
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉₇×#1801

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
prime radical	C₂³×C₉₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

272 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 117
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 117 points:

(2,4)(5,6,8,12,9,11,17,19)(7,13,14,20,15,10,16,18)(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);(1,2,3,4);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)

Conjugacy classes & character table

3104 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 533);