C₉₇×#2155 #27138113

order	12416
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#2155

Subgroups

	subgroup	G/H
center	C₄×C₉₇	#106
commutator	C₈	C₂²×C₄×C₉₇
Frattini, cosocle	C₂×C₈	C₂³×C₉₇
socle	C₂×C₉₇	C₂×#106
prime radical	C₂³×C₉₇	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ C₂²×C₈×C₉₇ ◁ G

144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,12,15,26)(4,9)(5,18)(6,21)(7,11)(8,14,23,28)(10,19)(13,25,16,30)(17,24)(20,31)(22,27,32,29)(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);(1,3,5,14,6,16,19,29,7,15,20,28,4,13,17,27)(2,8,10,12,9,22,18,25,11,23,24,26,21,32,31,30);(1,4,7,6)(2,9,11,21)(3,13,15,16)(5,17,20,19)(8,22,23,32)(10,18,24,31)(12,25,26,30)(14,27,28,29)

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, 887);