C₆₇×#7156 #30686407

order	17152
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#7156

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	#106
commutator	C₈	C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

234 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 95
transitive degree	μ_t(G) = 17152

A minimal faithful permutation representation — 2 generators on 95 points:

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

Conjugacy classes & character table

4288 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 144);