C₉₇×#1301 #27137259

order	12416
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#1301

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉₇	C₂²⋊C₄
commutator	C₂×C₄	C₄²×C₉₇
Frattini, cosocle	C₂×C₄²	C₂²×C₉₇
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

148 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 121
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 2 generators on 121 points:

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

Conjugacy classes & character table

3686 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₉₇×#7027, C₉₇×#7029, C₉₇×#7031, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 33);