C₇₁×#12380 #31833252

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₁×#12380

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₁
prime radical	C₇₁×#3391	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ C₈×Q₈×C₇₁ ◁ G

344 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 103
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 3 generators on 103 points:

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

Conjugacy classes & character table

2840 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5368);