C₉₇×#1373 #27137331

order	12416
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₉₇×#1373

Subgroups

	subgroup	G/H
center	C₈×C₉₇	D₈
commutator	C₈	C₂×C₈×C₉₇
Frattini, cosocle	C₄×C₈	C₂²×C₉₇
socle	C₂×C₉₇	#335
prime radical	C₂²×C₉₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₉₇ ◁ C₂×C₈×C₉₇ ◁ C₄×C₈×C₉₇ ◁ G

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 2 generators on 129 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₉₇×#7100

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 105);

C97×#1373 #27137331