C₆₇×#7503 #30686754

order	17152
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₇×#7503

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#354
commutator	C₄²	C₄²×C₆₇
Frattini, cosocle	#532	C₂²×C₆₇
socle	C₂×C₆₇	#1393
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #532×C₆₇ ◁ G

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 99 points:

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

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, 491);