C₉₇×#1374 #27137332

order	12416
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₉₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉₇×#1374

Subgroups

	subgroup	G/H
center	C₂²×C₈×C₉₇	C₂²
commutator	C₂	C₄×C₁₆×C₉₇
Frattini, cosocle	C₂²×C₈	C₂²×C₉₇
socle, prime radical	C₂³×C₉₇	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₉₇ ◁ G

160 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 121
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 2 generators on 121 points:

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

Conjugacy classes & character table

7760 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 106);