C₆₇×#13578 #30692829

order	17152
nilpotency class	5
isoclinism	isoclinic class of #13578
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#13578

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	C₂×#353
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	#124	C₂³×C₆₇
prime radical	C₆₇×#3153	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂³×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#124×C₆₇ ◁ G

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

normal subgroups only

This group has 420 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)(3,11,14,26)(4,15,16,31)(5,18,20,9)(6,10)(7,13,22,29)(8,17)(12,24,19,27)(21,28,23,30)(25,32)(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)(4,13)(5,19)(6,12)(8,24)(9,22)(10,21)(11,25)(14,20)(15,27)(16,28)(17,29)(18,23)(26,31)(30,32);(3,12)(7,21)(11,24)(13,28)(14,19)(22,23)(26,27)(29,30)

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, 6566);