C₉₇×#3483 #27139441

order	12416
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₉₇ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₉₇×#3483

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₉₇
socle	C₂²×C₉₇	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ G

1202 conjugacy classes of subgroups; only the 800 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 800 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 — 5 generators on 113 points:

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

Conjugacy classes & character table

4850 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 2215);