C₉₇×#2388 #27138346

order	12416
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₉₇×#2388

Subgroups

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

944 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 113
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 4 generators on 113 points:

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

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, 1120);