C₂×C₆₇×#1412 #30692844

order	17152
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₆₇×#1412

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#354
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	#127	C₂³×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#127×C₆₇ ◁ G

390 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 85
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 85 points:

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

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, 6581);