C₆₇×#13568 #30692819

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

Aliases

direct product

C₆₇×#13568

Subgroups

	subgroup	G/H
center	C₄×C₆₇	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₆₇
Frattini, cosocle	#124	C₂³×C₆₇
socle	C₂×C₆₇	C₂×AΣL(1,𝔽₁₆)
prime radical	C₆₇×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #532×C₆₇ ◁ G

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

normal subgroups only

This group has 498 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) = 83
transitive degree	μ_t(G) = 1072

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

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

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, 6556);