C₇₁×#11171 #31832043

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₁×#11171

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#381
commutator	C₂⁴	C₂²×C₄×C₇₁
Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
prime radical	C₂⁴×C₇₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ 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) = 107
transitive degree	μ_t(G) = 4544

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

(5,6,9,14,10,13,23,28)(7,15,17,29,18,11,21,25)(8,12,20,27,19,26,22,30)(16,31,33,36,32,24,34,35)(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);(1,2)(3,4)(5,7,10,18)(6,11,13,15)(8,16,19,32)(9,21,23,17)(12,24,26,31)(14,29,28,25)(20,34,22,33)(27,36,30,35);(2,4)(5,8)(6,12,13,26)(7,16)(9,22)(10,19)(11,24,15,31)(14,30,28,27)(17,34)(18,32)(20,23)(21,33)(25,36,29,35)

Conjugacy classes & character table

3266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 4159);