C₆₇×#7416 #30686667

order	17152
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#7416

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₄×C₆₇
Frattini, cosocle	#382	C₂²×C₆₇
prime radical	#537×C₆₇	C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #382×C₆₇ ◁ G

224 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 91
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 2 generators on 91 points:

(1,2)(3,5)(4,7)(6,8)(10,14,15,22,16,24,19,23)(11,17,18,20)(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);(2,5)(3,8)(6,7)(9,10)(11,14)(12,19)(13,16)(15,21)(17,22)(18,24)(20,23)

Conjugacy classes & character table

2077 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 404);