C₆₇×#9022 #30688273

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#9022

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

572 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 150 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:

(2,4)(5,6,9,13,10,14,8,12)(7,15,16,11,18,19,17,20)(21,22)(23,26)(24,31,32,27)(25,28)(29,33,35,36)(30,34)(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)(6,11)(8,16)(9,17)(10,18)(12,19)(13,15)(14,20)(21,23,25,30)(22,26,28,34)(24,29,32,35)(27,33,31,36);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,35)(34,36)

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, 2010);