C₆₇×#12062 #30691313

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#12062

Subgroups

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₆₇
Frattini, cosocle	C₄×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₈×D₄
prime radical	C₂×D₈×C₆₇	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

252 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 8576

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

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

Conjugacy classes & character table

5896 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 5050);