C₇₁×#7185 #31828057

order	18176
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#7185

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₁	#95
commutator	C₂³	C₄×C₈×C₇₁
Frattini, cosocle	C₂³×C₈	C₂²×C₇₁
socle	C₂²×C₇₁	#336
prime radical	C₂⁴×C₇₁	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂³×C₈×C₇₁ ◁ G

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

normal subgroups only

This group has 314 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) = 111
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 2 generators on 111 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,18,23,30,14,17,24,29,25,31,36,39)(11,19,21,32,22,33,35,40,20,15,26,27,34,28,37,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,17,19)(12,21)(13,22)(14,20)(16,27,29,32)(18,28,31,33)(23,35)(24,26)(25,34)(30,38,39,40)(36,37)

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, 173);