C₆₇×#11726 #30690977

order	17152
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₇×#11726

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	C₆₇×#2397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

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

normal subgroups only

This group has 606 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) = 83
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 83 points:

(2,6,7,8)(9,10)(11,14,16,13)(12,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(1,2)(3,6)(4,8)(5,7)(10,13)(11,16)(14,15);(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, 4714);