C₉₇×#1273 #27137231

order	12416
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₈²×C₉₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₉₇×#1273

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₉₇	C₂²
commutator	C₂	C₈²×C₉₇
Frattini, cosocle	C₂×C₄²	C₂²×C₉₇
socle, prime radical	C₂³×C₉₇	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₉₇ ◁ G

212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 2 generators on 117 points:

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

Conjugacy classes & character table

7760 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₉₇×#7014, C₉₇×#7015, C₉₇×#7016

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 5);

C97×#1273 #27137231