C₆₇×#33227 #30712478

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₆₇×#33227

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

986 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 318 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) = 95
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 4 generators on 95 points:

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

Conjugacy classes & character table

5092 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 26215);