C₉₇×#1915 #27137873

order	12416
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉₇×#1915

Subgroups

	subgroup	G/H
center, socle	C₂×C₉₇	#386
commutator	C₂³	C₂²×C₄×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
prime radical	C₂×#41×C₉₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ G

308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 129 points:

(1,2,5,10,6,9,11,24)(3,7,14,21,13,22,18,25)(4,15,17,26,16,8,19,23)(12,20,28,31,27,30,29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,3,11,14,6,13,5,18)(2,7,10,25,9,22,24,21)(4,12,17,29,16,27,19,28)(8,20,26,31,15,30,23,32);(1,4)(2,8)(3,12)(5,19)(6,16)(7,20)(9,15)(10,26)(11,17)(13,27)(14,29)(18,28)(21,32)(22,30)(23,24)(25,31)

Conjugacy classes & character table

2813 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 647);