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G = (C3×He3)⋊S3  order 486 = 2·35

2nd semidirect product of C3×He3 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C32⋊C9⋊4S3, (C3×He3)⋊2S3, C33.1(C3⋊S3), C3.1(He3⋊S3), C3.1(C33⋊S3), C32.24He3⋊3C2, C32.13(He3⋊C2), SmallGroup(486,43)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C32.24He3 — (C3×He3)⋊S3
Chief series
C1 — C3 — C32 — C33 — C3×He3 — C32.24He3 — (C3×He3)⋊S3
Lower central
C32.24He3 — (C3×He3)⋊S3
Upper central
C1

Generators and relations for (C3×He3)⋊S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, fbf=ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, dbd-1=bc-1, ebe-1=abc, cd=dc, ce=ec, fcf=c-1, ede-1=abd, fdf=d-1, fef=e-1 >

Subgroups: 1438 in 94 conjugacy classes, 13 normal (5 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, He3, C33, C33, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C3×He3, C32⋊D9, He3⋊4S3, C32.24He3, (C3×He3)⋊S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊S3, He3⋊S3, (C3×He3)⋊S3

Character table of (C3×He3)⋊S3

 class 123A3B3C3D3E3F3G3H3I3J3K3L6A6B9A9B9C9D9E9F
 size 1812222991818181818188181181818181818
ρ11111111111111111111111    trivial
ρ21-1111111111111-1-1111111    linear of order 2
ρ320222222-1-1-1-1-1-100-1-1222-1    orthogonal lifted from S3
ρ420222222-12-12-1200-1-1-1-1-1-1    orthogonal lifted from S3
ρ5202222222-12-12-100-1-1-1-1-1-1    orthogonal lifted from S3
ρ620222222-1-1-1-1-1-10022-1-1-12    orthogonal lifted from S3
ρ73-13333-3+3√-3/2-3-3√-3/2000000ζ65ζ6000000    complex lifted from He3⋊C2
ρ8313333-3+3√-3/2-3-3√-3/2000000ζ3ζ32000000    complex lifted from He3⋊C2
ρ93-13333-3-3√-3/2-3+3√-3/2000000ζ6ζ65000000    complex lifted from He3⋊C2
ρ10313333-3-3√-3/2-3+3√-3/2000000ζ32ζ3000000    complex lifted from He3⋊C2
ρ1160-3-3-36000-3000300000000    orthogonal lifted from C33⋊S3
ρ1260-3-3-360000030-300000000    orthogonal lifted from C33⋊S3
ρ1360-36-3-300-30300000000000    orthogonal lifted from C33⋊S3
ρ1460-36-3-30000-303000000000    orthogonal lifted from C33⋊S3
ρ1560-3-3-3600030-30000000000    orthogonal lifted from C33⋊S3
ρ1660-36-3-3003000-3000000000    orthogonal lifted from C33⋊S3
ρ1760-3-36-30000000000ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ920002ζ98-ζ94+ζ92+ζ9    orthogonal lifted from He3⋊S3
ρ18606-3-3-30000000000002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ920    orthogonal lifted from He3⋊S3
ρ19606-3-3-3000000000000ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ90    orthogonal lifted from He3⋊S3
ρ20606-3-3-3000000000000-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ90    orthogonal lifted from He3⋊S3
ρ2160-3-36-300000000002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9000-ζ98+2ζ97+ζ94+ζ92    orthogonal lifted from He3⋊S3
ρ2260-3-36-30000000000-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9000ζ95+2ζ94-ζ92+ζ9    orthogonal lifted from He3⋊S3

Smallest permutation representation of (C3×He3)⋊S3
►On 81 points
Generators in S81
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 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)

(1 2 3)(4 7 81)(5 8 79)(6 9 80)(10 11 12)(13 14 15)(16 20 24)(17 21 22)(18 19 23)(25 32 30)(26 33 28)(27 31 29)(34 39 41)(35 37 42)(36 38 40)(43 51 47)(44 49 48)(45 50 46)(52 54 53)(55 57 56)(58 60 59)(61 67 64)(62 68 65)(63 69 66)

(1 15 10)(2 13 11)(3 14 12)(4 7 81)(5 8 79)(6 9 80)(16 19 22)(17 20 23)(18 21 24)(25 28 31)(26 29 32)(27 30 33)(34 37 40)(35 38 41)(36 39 42)(43 46 49)(44 47 50)(45 48 51)(52 55 58)(53 56 59)(54 57 60)(61 64 67)(62 65 68)(63 66 69)(70 73 76)(71 74 77)(72 75 78)

(1 31 17)(2 32 18)(3 33 16)(4 75 69)(5 73 67)(6 74 68)(7 78 63)(8 76 61)(9 77 62)(10 28 23)(11 29 24)(12 30 22)(13 26 21)(14 27 19)(15 25 20)(34 55 46)(35 56 47)(36 57 48)(37 58 49)(38 59 50)(39 60 51)(40 52 43)(41 53 44)(42 54 45)(64 79 70)(65 80 71)(66 81 72)

(1 39 66)(2 37 64)(3 38 65)(4 30 57)(5 28 55)(6 29 56)(7 33 60)(8 31 58)(9 32 59)(10 36 63)(11 34 61)(12 35 62)(13 40 67)(14 41 68)(15 42 69)(16 43 70)(17 44 71)(18 45 72)(19 46 73)(20 47 74)(21 48 75)(22 49 76)(23 50 77)(24 51 78)(25 52 79)(26 53 80)(27 54 81)

(2 3)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 15)(11 14)(12 13)(16 32)(17 31)(18 33)(19 29)(20 28)(21 30)(22 26)(23 25)(24 27)(34 68)(35 67)(36 69)(37 65)(38 64)(39 66)(40 62)(41 61)(42 63)(49 80)(50 79)(51 81)(52 77)(53 76)(54 78)(55 74)(56 73)(57 75)(58 71)(59 70)(60 72)
 
G:=sub<Sym(81)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(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), (1,2,3)(4,7,81)(5,8,79)(6,9,80)(10,11,12)(13,14,15)(16,20,24)(17,21,22)(18,19,23)(25,32,30)(26,33,28)(27,31,29)(34,39,41)(35,37,42)(36,38,40)(43,51,47)(44,49,48)(45,50,46)(52,54,53)(55,57,56)(58,60,59)(61,67,64)(62,68,65)(63,69,66), (1,15,10)(2,13,11)(3,14,12)(4,7,81)(5,8,79)(6,9,80)(16,19,22)(17,20,23)(18,21,24)(25,28,31)(26,29,32)(27,30,33)(34,37,40)(35,38,41)(36,39,42)(43,46,49)(44,47,50)(45,48,51)(52,55,58)(53,56,59)(54,57,60)(61,64,67)(62,65,68)(63,66,69)(70,73,76)(71,74,77)(72,75,78), (1,31,17)(2,32,18)(3,33,16)(4,75,69)(5,73,67)(6,74,68)(7,78,63)(8,76,61)(9,77,62)(10,28,23)(11,29,24)(12,30,22)(13,26,21)(14,27,19)(15,25,20)(34,55,46)(35,56,47)(36,57,48)(37,58,49)(38,59,50)(39,60,51)(40,52,43)(41,53,44)(42,54,45)(64,79,70)(65,80,71)(66,81,72), (1,39,66)(2,37,64)(3,38,65)(4,30,57)(5,28,55)(6,29,56)(7,33,60)(8,31,58)(9,32,59)(10,36,63)(11,34,61)(12,35,62)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,46,73)(20,47,74)(21,48,75)(22,49,76)(23,50,77)(24,51,78)(25,52,79)(26,53,80)(27,54,81), (2,3)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,15)(11,14)(12,13)(16,32)(17,31)(18,33)(19,29)(20,28)(21,30)(22,26)(23,25)(24,27)(34,68)(35,67)(36,69)(37,65)(38,64)(39,66)(40,62)(41,61)(42,63)(49,80)(50,79)(51,81)(52,77)(53,76)(54,78)(55,74)(56,73)(57,75)(58,71)(59,70)(60,72)>;
 
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(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), (1,2,3)(4,7,81)(5,8,79)(6,9,80)(10,11,12)(13,14,15)(16,20,24)(17,21,22)(18,19,23)(25,32,30)(26,33,28)(27,31,29)(34,39,41)(35,37,42)(36,38,40)(43,51,47)(44,49,48)(45,50,46)(52,54,53)(55,57,56)(58,60,59)(61,67,64)(62,68,65)(63,69,66), (1,15,10)(2,13,11)(3,14,12)(4,7,81)(5,8,79)(6,9,80)(16,19,22)(17,20,23)(18,21,24)(25,28,31)(26,29,32)(27,30,33)(34,37,40)(35,38,41)(36,39,42)(43,46,49)(44,47,50)(45,48,51)(52,55,58)(53,56,59)(54,57,60)(61,64,67)(62,65,68)(63,66,69)(70,73,76)(71,74,77)(72,75,78), (1,31,17)(2,32,18)(3,33,16)(4,75,69)(5,73,67)(6,74,68)(7,78,63)(8,76,61)(9,77,62)(10,28,23)(11,29,24)(12,30,22)(13,26,21)(14,27,19)(15,25,20)(34,55,46)(35,56,47)(36,57,48)(37,58,49)(38,59,50)(39,60,51)(40,52,43)(41,53,44)(42,54,45)(64,79,70)(65,80,71)(66,81,72), (1,39,66)(2,37,64)(3,38,65)(4,30,57)(5,28,55)(6,29,56)(7,33,60)(8,31,58)(9,32,59)(10,36,63)(11,34,61)(12,35,62)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,46,73)(20,47,74)(21,48,75)(22,49,76)(23,50,77)(24,51,78)(25,52,79)(26,53,80)(27,54,81), (2,3)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,15)(11,14)(12,13)(16,32)(17,31)(18,33)(19,29)(20,28)(21,30)(22,26)(23,25)(24,27)(34,68)(35,67)(36,69)(37,65)(38,64)(39,66)(40,62)(41,61)(42,63)(49,80)(50,79)(51,81)(52,77)(53,76)(54,78)(55,74)(56,73)(57,75)(58,71)(59,70)(60,72) );
 
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(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)], [(1,2,3),(4,7,81),(5,8,79),(6,9,80),(10,11,12),(13,14,15),(16,20,24),(17,21,22),(18,19,23),(25,32,30),(26,33,28),(27,31,29),(34,39,41),(35,37,42),(36,38,40),(43,51,47),(44,49,48),(45,50,46),(52,54,53),(55,57,56),(58,60,59),(61,67,64),(62,68,65),(63,69,66)], [(1,15,10),(2,13,11),(3,14,12),(4,7,81),(5,8,79),(6,9,80),(16,19,22),(17,20,23),(18,21,24),(25,28,31),(26,29,32),(27,30,33),(34,37,40),(35,38,41),(36,39,42),(43,46,49),(44,47,50),(45,48,51),(52,55,58),(53,56,59),(54,57,60),(61,64,67),(62,65,68),(63,66,69),(70,73,76),(71,74,77),(72,75,78)], [(1,31,17),(2,32,18),(3,33,16),(4,75,69),(5,73,67),(6,74,68),(7,78,63),(8,76,61),(9,77,62),(10,28,23),(11,29,24),(12,30,22),(13,26,21),(14,27,19),(15,25,20),(34,55,46),(35,56,47),(36,57,48),(37,58,49),(38,59,50),(39,60,51),(40,52,43),(41,53,44),(42,54,45),(64,79,70),(65,80,71),(66,81,72)], [(1,39,66),(2,37,64),(3,38,65),(4,30,57),(5,28,55),(6,29,56),(7,33,60),(8,31,58),(9,32,59),(10,36,63),(11,34,61),(12,35,62),(13,40,67),(14,41,68),(15,42,69),(16,43,70),(17,44,71),(18,45,72),(19,46,73),(20,47,74),(21,48,75),(22,49,76),(23,50,77),(24,51,78),(25,52,79),(26,53,80),(27,54,81)], [(2,3),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,15),(11,14),(12,13),(16,32),(17,31),(18,33),(19,29),(20,28),(21,30),(22,26),(23,25),(24,27),(34,68),(35,67),(36,69),(37,65),(38,64),(39,66),(40,62),(41,61),(42,63),(49,80),(50,79),(51,81),(52,77),(53,76),(54,78),(55,74),(56,73),(57,75),(58,71),(59,70),(60,72)]])
 

Matrix representation of (C3×He3)⋊S3 ►in GL12(𝔽19)

1810000000000
1800000000000
0018100000000
0018000000000
7151001000000
15160181818000000
0000001810000
0000001800000
0000000018100
0000000018000
00000018101801
000000101811818
,
1810000000000
1800000000000
0001800000000
0011800000000
7150110000000
7150101000000
0000001810000
0000001800000
000000001000
000000000100
000000101821818
0000001810010
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
0000000180000
0000001180000
0000000001800
0000000011800
000000118011818
00000018011810
,
100000000000
010000000000
0001800000000
0011800000000
811801818000000
000110000000
000000001000
000000000100
0000000000181
0000002181821718
000000118011818
000000218011818
,
0000181000000
8118181718000000
100000000000
010000000000
1501212315000000
1501312315000000
0000000180000
0000001180000
000000001000
000000000100
0000001800010
0000001800001
,
010000000000
100000000000
8118181718000000
0000181000000
4150010000000
4150110000000
000000100000
0000001180000
0000000000181
00000017111712
00000018101811
00000018111811

G:=sub<GL(12,GF(19))| [18,18,0,0,7,15,0,0,0,0,0,0,1,0,0,0,15,16,0,0,0,0,0,0,0,0,18,18,1,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,18,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,18,18,0,18,0,0,0,0,0,0,0,0,1,0,18,1,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18],[18,18,0,0,7,7,0,0,0,0,0,0,1,0,0,0,15,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18,18,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,18,0,0,0,0,0,0,18,18,0,0,18,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,18,18,1,18,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[1,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,18,0,1,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,2,0,0,0,0,0,0,0,0,0,18,18,18,0,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,0,0,1,0,2,1,1,0,0,0,0,0,0,0,0,18,17,18,18,0,0,0,0,0,0,0,0,1,18,18,18],[0,8,1,0,15,15,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,18,0,0,12,13,0,0,0,0,0,0,0,18,0,0,12,12,0,0,0,0,0,0,18,17,0,0,3,3,0,0,0,0,0,0,1,18,0,0,15,15,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,18,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,1,8,0,4,4,0,0,0,0,0,0,1,0,1,0,15,15,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,0,0,0,0,0,0,17,18,1,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,17,18,18,0,0,0,0,0,0,0,18,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,17,18,18,0,0,0,0,0,0,0,0,18,1,1,1,0,0,0,0,0,0,0,0,1,2,1,1] >;
 

(C3×He3)⋊S3 in GAP, Magma, Sage, TeX 

(C_3\times {\rm He}_3)\rtimes S_3
 
% in TeX
 
G:=Group("(C3xHe3):S3");
 
// GroupNames label
 
G:=SmallGroup(486,43);
 
// by ID
 
G=gap.SmallGroup(486,43);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,224,3027,2817,1383,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,f*b*f=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=a*b*c,c*d=d*c,c*e=e*c,f*c*f=c^-1,e*d*e^-1=a*b*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

Export

Character table of (C3×He3)⋊S3 in TeX

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ℤ
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