C₇₁×#11957 #31832829

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₁×#11957

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₄×C₈	C₂³×C₇₁
prime radical	C₂²×D₈×C₇₁	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

406 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 111
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 111 points:

(1,2,3,6,4,5,7,8)(9,10,13,18,14,19,28,36,15,20,29,37,27,35,12,17)(11,21,23,33,24,34,22,38,25,39,26,16,30,31,40,32)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,5)(3,7)(9,11)(10,16)(12,26)(13,22)(14,30)(15,25)(17,34)(18,21)(19,38)(20,33)(23,28)(24,27)(29,40)(31,36)(32,35)(37,39);(2,5)(6,8)

Conjugacy classes & character table

3692 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 4945);