C₂×#441×C₉₇ #27138899

order	12416
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₉₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#441×C₉₇

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₉₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₉₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

480 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 119
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 4 generators on 119 points:

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

Conjugacy classes & character table

4268 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 1673);