C₆₇×#12864 #30692115

order	17152
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#12864

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×#41	C₂³×C₆₇
socle	C₂×C₆₇	#1820
prime radical	C₂×#41×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×#41×C₆₇ ◁ G

342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,12,15,10)(4,16)(5,9)(6,11)(7,14,21,28)(8,13,23,27)(17,20)(18,25)(19,24)(22,26,31,32)(29,30)(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)(2,7,11,21)(4,13)(5,14)(6,15)(8,20,23,30)(9,12,24,10)(16,22,25,31)(17,26)(18,27)(19,28)(29,32);(1,4,5,17,6,18,19,29)(2,8,9,22,11,23,24,31)(3,13,14,26,15,27,28,32)(7,20,12,25,21,30,10,16)

Conjugacy classes & character table

2680 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 5852);