C₉₇×#1389 #27137347

order	12416
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#1389

Subgroups

	subgroup	G/H
center	C₄×C₉₇	#102
commutator	C₈	C₄²×C₉₇
Frattini, cosocle	C₂²×C₈	C₂²×C₉₇
socle	C₂×C₉₇	#340
prime radical	C₂³×C₉₇	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ C₂²×C₈×C₉₇ ◁ G

128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 2 generators on 129 points:

(1,2,5,10,7,11,6,22)(3,12,14,29,17,25,16,31)(4,18,19,27,21,32,20,9)(8,23,24,15,26,30,28,13)(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);(1,3,4,15,5,14,19,30,7,17,21,13,6,16,20,23)(2,8,9,25,22,28,32,29,11,26,27,12,10,24,18,31)

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, 121);