C₆₇×#11168 #30690419

order	17152
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₇×#11168

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#381
commutator	C₂⁴	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

802 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 103
transitive degree	μ_t(G) = 4288

A minimal faithful permutation representation — 3 generators on 103 points:

(5,6,9,14,10,13,23,28)(7,15,17,29,18,11,22,25)(8,19,21,33,20,12,26,27)(16,24,32,35,31,30,34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,2)(3,4)(5,7,10,18)(6,11,13,15)(8,16,20,31)(9,22,23,17)(12,24,19,30)(14,29,28,25)(21,34,26,32)(27,36,33,35);(2,4)(5,8)(6,12,13,19)(7,16)(9,21)(10,20)(11,24,15,30)(14,27,28,33)(17,32)(18,31)(22,34)(23,26)(25,35,29,36)

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, 4156);