C₆₇×#11431 #30690682

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#11431

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	#99
commutator	C₂×C₄	C₂²×C₈×C₆₇
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#99
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

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

normal subgroups only

This group has 602 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) = 91
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 91 points:

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

Conjugacy classes & character table

3484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 4419);