C₇₁×#13697 #31834569

order	18176
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₁×#13697

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₁
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂²×C₈×C₇₁ ◁ C₂²×C₁₆×C₇₁ ◁ G

210 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 107
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 3 generators on 107 points:

(1,2)(3,4)(6,11)(7,12)(8,16)(9,19)(13,21)(14,30)(15,31)(17,33)(18,27)(20,32)(22,23)(24,29)(25,26)(28,36)(34,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(5,6,12,23,8,14,25,31,19,29,33,36,32,34,18,21,10,13,27,35,20,28,17,24,9,15,26,30,16,22,7,11);(2,4)(6,13)(11,21)(14,28)(15,29)(22,34)(23,35)(24,31)(30,36)

Conjugacy classes & character table

3266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 6685);