C₉₇×#1275 #27137233

order	12416
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₉₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉₇×#1275

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉₇	C₄²
commutator	C₄	C₄×C₈×C₉₇
Frattini, cosocle	C₂×C₄²	C₂²×C₉₇
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ G

152 conjugacy classes of subgroups.

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:

(1,2,3,5,4,6,7,8)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,19,23,21)(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);(9,11,13,21,14,20,12,19)(10,15,16,18,17,23,22,24)

Conjugacy classes & character table

4268 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₉₇×#7014, C₉₇×#7019, C₉₇×#7020, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 7);

C97×#1275 #27137233