C₇₁×#12269 #31833141

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#12269

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇₁
prime radical	C₂⁴×C₇₁	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₇₁ ◁ C₂×C₂²⋊C₄×C₇₁ ◁ G

372 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 4544

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

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

Conjugacy classes & character table

1988 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5257);