C₇₁×#13671 #31834543

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

Aliases

direct product

C₇₁×#13671

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₁	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₁
prime radical	C₇₁×#3415	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ #131×C₇₁ ◁ #504×C₇₁ ◁ G

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

normal subgroups only

This group has 284 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) = 135
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 135 points:

(1,2,8,15)(3,16,20,45)(4,21,27,54)(5,14,7,12)(6,30,32,46)(9,33,37,17)(10,28,39,18)(11,40,44,25)(13,29,48,23)(19,51,52,38)(22,47,57,64)(24,34,26,53)(31,41,61,63)(35,50,49,56)(36,60,62,55)(42,59,43,58)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,3)(2,9)(4,22)(5,18)(6,31)(7,28)(8,20)(10,34)(11,41)(12,35)(13,47)(14,49)(15,37)(16,21)(17,25)(19,23)(24,55)(26,60)(27,57)(29,52)(30,36)(32,61)(33,40)(38,43)(39,53)(42,51)(44,63)(45,54)(46,62)(48,64)(50,59)(56,58);(1,3)(2,10)(4,17)(5,28)(6,19)(7,18)(8,20)(9,34)(11,36)(12,45)(13,38)(14,16)(15,39)(21,49)(22,25)(23,31)(24,56)(26,50)(27,33)(29,61)(30,41)(32,52)(35,54)(37,53)(40,57)(42,64)(43,47)(44,62)(46,63)(48,51)(55,58)(59,60)

Conjugacy classes & character table

2627 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 6659);