C₉₇×#2180 #27138138

order	12416
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#2180

Subgroups

	subgroup	G/H
center, socle	C₂×C₉₇	#442
commutator	C₈	C₂²×C₄×C₉₇
Frattini, cosocle	C₂×C₈	C₂³×C₉₇
prime radical	#135×C₉₇	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉₇ ◁ C₂×C₄×C₉₇ ◁ #97×C₉₇ ◁ G

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

normal subgroups only

This group has 142 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) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 129 points:

(1,2)(3,13,15,24)(4,17)(5,9)(6,21)(7,12)(8,14,25,31)(10,18)(11,23)(16,28,22,30)(19,29,32,26)(20,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,3)(2,8)(4,14)(5,19)(6,22)(7,15)(9,24)(10,28)(11,29)(12,25)(13,27)(16,23)(17,30)(18,31)(20,32)(21,26);(1,4,6,5,7,18,23,20)(2,9,11,17,12,27,21,10)(3,14,22,19,15,31,16,32)(8,24,29,30,25,13,26,28)

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, 912);