C₆₇×#13569 #30692820

order	17152
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₇×#13569

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₆₇
Frattini, cosocle	#124	C₂³×C₆₇
prime radical	C₂×#454×C₆₇	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂³×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#124×C₆₇ ◁ G

492 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 99
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 99 points:

(1,2)(3,11,14,29)(4,16,18,28)(5,20,22,9)(6,10)(7,13,24,31)(8,19)(12,26,15,30)(17,25,32,23)(21,27)(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);(1,3)(2,7)(4,17)(5,14)(6,12)(8,26)(9,24)(10,23)(11,27)(13,18)(15,22)(16,30)(19,31)(20,25)(21,32)(28,29);(3,12)(7,23)(11,26)(13,17)(14,15)(24,25)(29,30)(31,32)

Conjugacy classes & character table

1876 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6557);