C₉₇×#1335 #27137293

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

Aliases

direct product

C₉₇×#1335

Subgroups

	subgroup	G/H
center	C₈×C₉₇	D₈
commutator	C₈	C₂×C₈×C₉₇
Frattini, cosocle	C₄×C₈	C₂²×C₉₇
socle	C₂×C₉₇	D₄⋊C₈
prime radical	#135×C₉₇	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₉₇ ◁ C₂×C₈×C₉₇ ◁ C₄×C₈×C₉₇ ◁ G

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 2 generators on 113 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₉₇×#7068, C₉₇×#7069, C₉₇×#7074, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 67);

C97×#1335 #27137293