C₉₇×#1500 #27137458

order	12416
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉₇×#1500

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	#121×C₉₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 3104

A minimal faithful permutation representation — 3 generators on 121 points:

(2,5,6,7)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(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)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 232);