C₉₇×#1283 #27137241

order	12416
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#1283

Subgroups

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

160 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 6208

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

(1,2,4,7,6,8,5,3)(9,10,11,14,12,13,15,16)(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);(1,3,5,8,6,7,4,2)(10,13)(11,15)

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₉₇×#7016, C₉₇×#7022, C₉₇×#7023, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 15);