C₂×#342×C₉₇ #27137697

order	12416
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₉₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#342×C₉₇

Subgroups

	subgroup	G/H
center, socle	C₂³×C₉₇	C₂²⋊C₄
commutator	C₂²	C₂×C₄²×C₉₇
Frattini, cosocle	C₂⁴	C₂³×C₉₇
prime radical	C₂⁵×C₉₇	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂⁵×C₉₇ ◁ G

680 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 111
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 111 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 471);