C₉₇×#2175 #27138133

order	12416
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#2175

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂×D₈
commutator	C₈	C₂²×C₄×C₉₇
Frattini, cosocle	C₂×C₈	C₂³×C₉₇
prime radical	C₂×D₈×C₉₇	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ C₄×C₈×C₉₇ ◁ G

160 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 6208

A minimal faithful permutation representation — 3 generators on 117 points:

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

Conjugacy classes & character table

3104 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 907);