C₂×B(4,𝔽₂)×C₉₇ #27138981

order	12416
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉₇ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×B(4,𝔽₂)×C₉₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₉₇
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂⁴×C₉₇ ◁ G

1018 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 107
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 4 generators on 107 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);(104,107);(100,101)(102,104)(103,105)(106,107);(100,102)(101,104)(103,106)(105,107)

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, 1755);