C₉₇×#1394 #27137352

order	12416
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#1394

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	#99
commutator	C₂×C₄	C₄²×C₉₇
Frattini, cosocle	C₂²×Q₈	C₂²×C₉₇
prime radical	C₂³×C₉₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂³×C₉₇ ◁ C₂²×Q₈×C₉₇ ◁ G

172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 117
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 2 generators on 117 points:

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

Conjugacy classes & character table

2522 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 126);