C₉₇×#2204 #27138162

order	12416
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉₇×#2204

Subgroups

	subgroup	G/H
center, socle	C₂×C₉₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³×C₉₇
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉₇ ◁ C₂²×C₉₇ ◁ C₂×Q₈×C₉₇ ◁ G

236 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 1552

A minimal faithful permutation representation — 3 generators on 113 points:

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

Conjugacy classes & character table

1940 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 936);