C₉₇×#1315 #27137273

order	12416
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉₇×#1315

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₈×C₉₇
Frattini, cosocle	C₂²×C₈	C₂²×C₉₇
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂×#41×C₉₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ C₂²×C₈×C₉₇ ◁ G

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 2 generators on 137 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,19,28,36,15,20,29,37,12,17,26,34)(11,21,23,38,22,32,39,35,24,33,27,40,25,16,30,31)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,33)(18,35)(19,21)(20,32)(23,29)(26,39)(28,30)(31,37)(34,38)(36,40)

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₉₇×#7058, C₉₇×#7059, C₉₇×#7060, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 47);

C97×#1315 #27137273