C₇₁×#8822 #31829694

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₁×#8822

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₁	#127
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	#384
prime radical	C₂³×C₇₁	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

354 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 214 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) = 95
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 3 generators on 95 points:

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

Conjugacy classes & character table

5396 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 1810);