C₆₇×#7504 #30686755

order	17152
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#7504

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#354
commutator	C₄²	C₄²×C₆₇
Frattini, cosocle	C₂×#127	C₂²×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

336 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 87
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 2 generators on 87 points:

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

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, 492);