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## G = 3- 1+2.Dic3order 324 = 22·34

### The non-split extension by 3- 1+2 of Dic3 acting via Dic3/C2=S3

Aliases: 3- 1+2.Dic3, C3.He3⋊C4, (C3×C18).9S3, (C3×C9).2Dic3, C6.5(He3⋊C2), C3.5(He33C4), C32.4(C3⋊Dic3), (C2×3- 1+2).3S3, C2.(3- 1+2.S3), (C3×C6).4(C3⋊S3), (C2×C3.He3).C2, SmallGroup(324,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3.He3 — 3- 1+2.Dic3
 Chief series C1 — C3 — C32 — C3×C9 — C3.He3 — C2×C3.He3 — 3- 1+2.Dic3
 Lower central C3.He3 — 3- 1+2.Dic3
 Upper central C1 — C2

Generators and relations for 3- 1+2.Dic3
G = < a,b,c,d | a9=b3=1, c6=a6, d2=a6c3, bab-1=a4, cac-1=a4b-1, dad-1=a-1, bc=cb, bd=db, dcd-1=a3c5 >

Character table of 3- 1+2.Dic3

 class 1 2 3A 3B 3C 4A 4B 6A 6B 6C 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A 18B 18C 18D 18E 18F size 1 1 2 3 3 27 27 2 3 3 6 6 6 18 18 18 27 27 27 27 6 6 6 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 1 1 -i i -1 -1 -1 1 1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 1 1 i -i -1 -1 -1 1 1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 2 2 2 0 0 2 2 2 -1 -1 -1 -1 2 -1 0 0 0 0 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ6 2 2 2 2 2 0 0 2 2 2 -1 -1 -1 2 -1 -1 0 0 0 0 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 2 0 0 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 2 2 2 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 2 2 2 0 0 2 2 2 -1 -1 -1 -1 -1 2 0 0 0 0 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ9 2 -2 2 2 2 0 0 -2 -2 -2 -1 -1 -1 -1 2 -1 0 0 0 0 1 1 1 1 -2 1 symplectic lifted from Dic3, Schur index 2 ρ10 2 -2 2 2 2 0 0 -2 -2 -2 -1 -1 -1 2 -1 -1 0 0 0 0 1 1 1 -2 1 1 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 2 2 2 0 0 -2 -2 -2 2 2 2 -1 -1 -1 0 0 0 0 -2 -2 -2 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 2 2 2 0 0 -2 -2 -2 -1 -1 -1 -1 -1 2 0 0 0 0 1 1 1 1 1 -2 symplectic lifted from Dic3, Schur index 2 ρ13 3 3 3 -3-3√-3/2 -3+3√-3/2 1 1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ32 ζ3 ζ3 ζ32 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ14 3 3 3 -3-3√-3/2 -3+3√-3/2 -1 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ65 ζ65 ζ6 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ15 3 3 3 -3+3√-3/2 -3-3√-3/2 1 1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ3 ζ32 ζ32 ζ3 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ16 3 3 3 -3+3√-3/2 -3-3√-3/2 -1 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ6 ζ6 ζ65 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ17 3 -3 3 -3+3√-3/2 -3-3√-3/2 i -i -3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 0 0 0 0 0 0 complex lifted from He3⋊3C4 ρ18 3 -3 3 -3+3√-3/2 -3-3√-3/2 -i i -3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 0 0 0 0 0 0 complex lifted from He3⋊3C4 ρ19 3 -3 3 -3-3√-3/2 -3+3√-3/2 i -i -3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 0 0 0 0 0 0 complex lifted from He3⋊3C4 ρ20 3 -3 3 -3-3√-3/2 -3+3√-3/2 -i i -3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 0 0 0 0 0 0 complex lifted from He3⋊3C4 ρ21 6 6 -3 0 0 0 0 -3 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 orthogonal lifted from 3- 1+2.S3 ρ22 6 6 -3 0 0 0 0 -3 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 orthogonal lifted from 3- 1+2.S3 ρ23 6 6 -3 0 0 0 0 -3 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 orthogonal lifted from 3- 1+2.S3 ρ24 6 -6 -3 0 0 0 0 3 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 symplectic faithful, Schur index 2 ρ25 6 -6 -3 0 0 0 0 3 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 -3 0 0 0 0 3 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of 3- 1+2.Dic3
On 108 points
Generators in S108
(1 20 43 13 32 37 7 26 49)(2 33 44 14 27 38 8 21 50)(3 28 45 15 22 39 9 34 51)(4 23 46 16 35 40 10 29 52)(5 36 47 17 30 41 11 24 53)(6 31 48 18 25 42 12 19 54)(55 73 105 67 85 99 61 79 93)(56 74 100 68 86 94 62 80 106)(57 75 95 69 87 107 63 81 101)(58 76 108 70 88 102 64 82 96)(59 77 103 71 89 97 65 83 91)(60 78 98 72 90 92 66 84 104)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)
(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 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)
(1 62 10 71)(2 61 11 70)(3 60 12 69)(4 59 13 68)(5 58 14 67)(6 57 15 66)(7 56 16 65)(8 55 17 64)(9 72 18 63)(19 95 28 104)(20 94 29 103)(21 93 30 102)(22 92 31 101)(23 91 32 100)(24 108 33 99)(25 107 34 98)(26 106 35 97)(27 105 36 96)(37 74 46 83)(38 73 47 82)(39 90 48 81)(40 89 49 80)(41 88 50 79)(42 87 51 78)(43 86 52 77)(44 85 53 76)(45 84 54 75)

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

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

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

Matrix representation of 3- 1+2.Dic3 in GL6(𝔽37)

 0 0 36 36 0 0 0 0 1 0 0 0 0 0 0 0 36 36 0 0 0 0 1 0 0 1 0 0 0 0 36 36 0 0 0 0
,
 36 36 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 36 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 31 0 0 0 0 6 17 0 0 0 0 0 0 11 31 0 0 0 0 6 17 0 0 0 0 0 0 6 17 0 0 0 0 20 26
,
 0 0 7 14 0 0 0 0 7 30 0 0 7 14 0 0 0 0 7 30 0 0 0 0 0 0 0 0 23 30 0 0 0 0 7 14

G:=sub<GL(6,GF(37))| [0,0,0,0,0,36,0,0,0,0,1,36,36,1,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0],[36,1,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,6,0,0,0,0,31,17,0,0,0,0,0,0,11,6,0,0,0,0,31,17,0,0,0,0,0,0,6,20,0,0,0,0,17,26],[0,0,7,7,0,0,0,0,14,30,0,0,7,7,0,0,0,0,14,30,0,0,0,0,0,0,0,0,23,7,0,0,0,0,30,14] >;

3- 1+2.Dic3 in GAP, Magma, Sage, TeX

3_-^{1+2}.{\rm Dic}_3
% in TeX

G:=Group("ES-(3,1).Dic3");
// GroupNames label

G:=SmallGroup(324,25);
// by ID

G=gap.SmallGroup(324,25);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,2090,986,3171,303,453,7564,1096,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=1,c^6=a^6,d^2=a^6*c^3,b*a*b^-1=a^4,c*a*c^-1=a^4*b^-1,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^3*c^5>;
// generators/relations

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