C₇₁×#12897 #31833769

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#12897

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₁	#117
commutator	C₂×C₄	C₂²×C₈×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
socle	C₂²×C₇₁	#432
prime radical	C₂³×C₇₁	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×#97×C₇₁ ◁ G

278 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,5)(4,6)(7,8)(9,10,14,20,13,19,29,36,15,21,31,17,30,18,12,27)(11,22,25,37,24,16,23,32,26,33,38,40,39,34,28,35)(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);(1,3,4,7)(2,5,6,8)(9,11,15,26)(10,16)(12,28,29,23)(13,24,30,39)(14,25,31,38)(17,35)(18,22)(19,33)(20,32)(21,34)(27,37)(36,40);(2,6)(5,8)(9,12,13,14,15,29,30,31)(10,17,18,36,21,20,19,27)(11,23,24,38,26,28,39,25)(16,32,22,37,34,35,33,40)

Conjugacy classes & character table

4544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5885);