C₆₇×#8133 #30687384

order	17152
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₆₇ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₆₇×#8133

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₆₇	C₂³
commutator	C₂²	C₂×C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle, prime radical	C₂³×C₆₇	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₇ ◁ G

446 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 17152

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

(1,2)(3,6)(4,5)(7,8)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(2,5)(6,8)(9,10,11,13,12,14,15,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,3,4,7)(2,6,5,8)(19,25)(22,28)(26,31)(29,32)

Conjugacy classes & character table

7504 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 1121);