C₉₇×#1735 #27137693

order	12416
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₉₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₉₇×#1735

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₉₇	D₄
commutator	C₄	C₂×C₄²×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
socle	C₂³×C₉₇	C₂×D₄
prime radical	C₂⁴×C₉₇	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉₇ ◁ C₂³×C₄×C₉₇ ◁ G

424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 6208

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 467);