C₉₇×#1612 #27137570

order	12416
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#1612

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
prime radical	#538×C₉₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ G

232 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 121
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

2231 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 344);