C₆₇×#7255 #30686506

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#7255

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#327
commutator	C₂×C₈	C₄²×C₆₇
Frattini, cosocle	C₂²×M₁₆	C₂²×C₆₇
prime radical	C₂⁴×C₆₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

324 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 4288

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

(1,2,3,4)(6,11)(7,15)(8,9)(12,16)(13,20)(14,19)(17,18)(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);(5,6,8,12,10,14,9,13)(7,16,17,19,18,20,15,11)

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, 243);