C₆₇×#13560 #30692811

order	17152
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#13560

Subgroups

	subgroup	G/H
center	C₄×C₆₇	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₆₇
Frattini, cosocle	#127	C₂³×C₆₇
socle	C₂×C₆₇	C₂×AΣL(1,𝔽₁₆)
prime radical	C₆₇×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #532×C₆₇ ◁ G

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

normal subgroups only

This group has 486 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)(3,12,14,7)(4,8)(5,10,19,9)(6,11)(13,26,27,20)(15,25,18,21)(16,23,30,22)(17,24)(28,32,29,31)(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)(4,13)(5,18)(6,15)(8,20)(9,12)(10,25)(11,21)(14,19)(16,29)(17,28)(22,26)(23,32)(24,31)(27,30);(1,4,6,17)(2,8,11,24)(3,13,15,28)(5,16,19,30)(7,20,21,31)(9,22,10,23)(12,26,25,32)(14,27,18,29)

Conjugacy classes & character table

2278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6548);