C₆₇×#11745 #30690996

order	17152
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₇×#11745

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
prime radical	C₂³×C₆₇	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

346 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 4288

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

(2,6,7,8)(10,15,16,23)(11,19)(12,13)(17,20,18,24)(21,22)(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);(1,2)(3,6)(4,8)(5,7)(9,10,14,16)(11,20,22,24)(12,18,13,17)(15,19,23,21);(1,3,5,4)(2,6,7,8)(10,16)(15,23)(17,18)(20,24)

Conjugacy classes & character table

3082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 4733);