C₇₁×#7192 #31828064

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#7192

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₁	#99
commutator	C₂³	C₄×C₈×C₇₁
Frattini, cosocle	C₂²×M₁₆	C₂²×C₇₁
socle	C₂²×C₇₁	#324
prime radical	C₂⁴×C₇₁	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂²×M₁₆×C₇₁ ◁ G

310 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 2 generators on 107 points:

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

Conjugacy classes & character table

3692 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 180);