C₇₁×#13333 #31834205

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₁×#13333

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
prime radical	C₂×#135×C₇₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

546 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 107
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 107 points:

(1,2)(3,4)(5,6)(7,18)(8,24)(9,27)(10,16)(11,30)(12,17)(13,20)(14,23)(15,19)(21,33)(22,32)(25,31)(26,35)(28,34)(29,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(2,4)(5,7,12,23)(6,13,17,31)(8,9,26,29)(10,22,11,21)(14,27,18,36)(15,33,34,32)(16,35,30,24)(19,20,28,25);(5,8,12,26)(6,14,17,18)(7,19,23,28)(9,21,29,22)(10,20,11,25)(13,15,31,34)(16,33,30,32)(24,36,35,27)

Conjugacy classes & character table

2840 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 6321);