C₉₇×#1279 #27137237

order	12416
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₉₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉₇×#1279

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₉₇	D₄
commutator	C₄	C₄×C₈×C₉₇
Frattini, cosocle	C₂×C₄²	C₂²×C₉₇
socle, prime radical	C₂³×C₉₇	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

208 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 117
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 2 generators on 117 points:

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

Conjugacy classes & character table

5432 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₉₇×#7015, C₉₇×#7022, C₉₇×#7024, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 11);