C₉₇×#1805 #27137763

order	12416
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉₇×#1805

Subgroups

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

292 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 3 generators on 121 points:

(1,2,3,4)(6,8)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23)(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);(5,6,7,8)(10,14)(15,23)(16,21)(17,24);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)

Conjugacy classes & character table

3686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 537);