C₇₁×#13596 #31834468

order	18176
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₁×#13596

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₁	C₂×#354
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	#127	C₂³×C₇₁
prime radical	#532×C₇₁	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ #532×C₇₁ ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 2272

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

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

Conjugacy classes & character table

1988 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 6584);