C₆₇×#8316 #30687567

order	17152
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₆₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₆₇×#8316

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	#120
commutator	C₂³	C₂²×C₈×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	#379
prime radical	C₂×H(5,𝔽₂)×C₆₇	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

1118 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 83
transitive degree	μ_t(G) = 4288

A minimal faithful permutation representation — 3 generators on 83 points:

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

Conjugacy classes & character table

4288 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 1304);