C₉₇×#2178 #27138136

order	12416
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#2178

Subgroups

	subgroup	G/H
center	C₈×C₉₇	D₈
commutator	C₈	C₂²×C₄×C₉₇
Frattini, cosocle	C₂×C₈	C₂³×C₉₇
socle	C₂×C₉₇	C₄×D₈
prime radical	#508×C₉₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₉₇ ◁ C₂×C₈×C₉₇ ◁ C₄×C₈×C₉₇ ◁ G

146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 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:

(1,2,6,10,7,12,22,11)(3,13,15,24,17,30,16,32)(4,9,18,27,19,28,5,20)(8,23,25,31,26,21,29,14)(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)(2,8)(4,14)(5,21)(6,15)(7,17)(9,24)(10,25)(11,29)(12,26)(13,20)(16,22)(18,23)(19,31)(27,30)(28,32);(1,4,6,18,7,19,22,5)(2,9,10,27,12,28,11,20)(3,14,15,23,17,31,16,21)(8,24,25,30,26,32,29,13)

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, 910);