C₇₁×#12264 #31833136

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₁×#12264

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇₁
prime radical	C₂⁵×C₇₁	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₇₁ ◁ C₂×C₂²⋊C₄×C₇₁ ◁ G

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

normal subgroups only

This group has 468 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) = 2272

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

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

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, 5252);