C₂×#489×C₉₇ #27139090

order	12416
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#489×C₉₇

Subgroups

	subgroup	G/H
center, socle	C₂³×C₉₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
prime radical	C₂²×D₄×C₉₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

436 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 115
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 4 generators on 115 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,18);(3,5,6,9)(4,7,8,10)(11,13,15,12,16,17,14,18);(4,8)(7,10)(11,13,14,18,16,17,15,12)

Conjugacy classes & character table

3104 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 1864);