C₂×C₇₁×#1403 #31834435

order	18176
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₇₁×#1403

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₇₁
Frattini, cosocle	#124	C₂³×C₇₁
prime radical	C₂×H(5,𝔽₂)×C₇₁	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×#124×C₇₁ ◁ G

514 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 89
transitive degree	μ_t(G) = 2272

A minimal faithful permutation representation — 3 generators on 89 points:

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

Conjugacy classes & character table

2414 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 6551);