C₆₇×#11728 #30690979

order	17152
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#11728

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	#99
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#99
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

470 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 106 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)(9,10)(11,17,19,24)(12,20,22,15)(13,16)(14,18)(21,23)(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);(1,2)(3,6)(4,8)(5,7)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3,5,4)(2,6,7,8)

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, 4716);