C₉₇×#2189 #27138147

order	12416
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉₇×#2189

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₉₇
prime radical	C₂×#135×C₉₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ C₂²×C₈×C₉₇ ◁ G

220 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 133
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 133 points:

(1,2)(3,4)(5,6,11,15)(7,16,19,34)(8,13,10,25)(9,21,24,35)(12,27,30,18)(14,20,32,23)(17,33,26,31)(22,28,36,29)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,28)(14,31)(15,30)(16,21)(18,23)(20,27)(24,36)(25,29)(32,33)(34,35);(2,4)(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,24,23)(12,28,30,29)(14,21,32,35)(16,33,34,31)(18,22,27,36)

Conjugacy classes & character table

2813 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 921);