C₉₇×#1350 #27137308

order	12416
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉₇×#1350

Subgroups

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

104 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 137
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 2 generators on 137 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,18,15,20,14,19)(11,21,23,34,25,38,24,37)(12,26,28,17,30,35,29,36)(16,27,31,22,33,39,32,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,20,33)(12,27,30,39)(13,24,14,23)(17,34,36,37)(18,32,19,31)(21,26,38,35)(22,28,40,29)

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₉₇×#7081, C₉₇×#7082, C₉₇×#7085, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 82);