C₉₇×#1305 #27137263

order	12416
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#1305

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	#95
commutator	C₂³	C₄²×C₉₇
Frattini, cosocle	C₂³×C₄	C₂²×C₉₇
prime radical	C₂⁴×C₉₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂³×C₄×C₉₇ ◁ G

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

normal subgroups only

This group has 200 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) = 113
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 2 generators on 113 points:

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

Conjugacy classes & character table

2522 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₉₇×#7032, C₉₇×#7033, C₉₇×#7034, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 37);