C₉₇×#1328 #27137286

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

Aliases

direct product

C₉₇×#1328

Subgroups

	subgroup	G/H
center	C₄×C₉₇	#99
commutator	C₂³	C₂×C₈×C₉₇
Frattini, cosocle	C₂×M₁₆	C₂²×C₉₇
socle	C₂×C₉₇	#324
prime radical	C₂³×C₉₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉₇ ◁ C₂×C₄×C₉₇ ◁ C₂×M₁₆×C₉₇ ◁ G

90 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:

(1,2,4,9,5,10,15,23,6,8,14,20,16,21,27,30)(3,7,12,17,13,19,26,28,11,18,24,22,25,29,32,31)(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)(2,7,8,18)(4,12)(5,13)(6,11)(9,22,20,17)(10,19,21,29)(14,24)(15,26)(16,25)(23,31,30,28)(27,32)

Conjugacy classes & character table

2522 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₉₇×#7064, C₉₇×#7065, C₉₇×#7066

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 60);

C97×#1328 #27137286