C₆₇×#12350 #30691601

order	17152
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#12350

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₇
prime radical	C₆₇×#3355	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ #436×C₆₇ ◁ G

298 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 107
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 3 generators on 107 points:

(2,3)(4,5)(7,8)(9,10,14,18)(11,19,21,38)(12,23,25,37)(13,28,26,17)(15,30,31,20)(16,33,35,27)(22,39,29,34)(24,36,40,32)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,2,6,8)(3,4,7,5)(9,11,14,21)(10,15,18,31)(12,24,25,40)(13,29,26,22)(16,34,35,39)(17,36,28,32)(19,37,38,23)(20,33,30,27);(1,3)(2,4)(5,8)(6,7)(12,25)(16,35)(19,38)(20,30)(23,37)(24,40)(27,33)(34,39)

Conjugacy classes & character table

2077 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 5338);