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## G = He3.Dic3order 324 = 22·34

### 1st non-split extension by He3 of Dic3 acting via Dic3/C2=S3

Aliases: He3.1Dic3, (C3×C9)⋊2C12, (C3×C18).2C6, C9⋊Dic32C3, He3.C32C4, (C2×He3).2S3, C6.3(C32⋊C6), C3.3(C32⋊C12), C2.(He3.S3), C32.7(C3×Dic3), (C3×C6).14(C3×S3), (C2×He3.C3).2C2, SmallGroup(324,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — He3.Dic3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×He3.C3 — He3.Dic3
 Lower central C3×C9 — He3.Dic3
 Upper central C1 — C2

Generators and relations for He3.Dic3
G = < a,b,c,d,e | a3=b3=c3=1, d6=ebe-1=b-1, e2=b-1d3, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, dcd-1=ab-1c, ce=ec, ede-1=bd5 >

Character table of He3.Dic3

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 9A 9B 9C 9D 9E 12A 12B 12C 12D 18A 18B 18C 18D 18E size 1 1 2 6 9 9 27 27 2 6 9 9 6 6 6 18 18 27 27 27 27 6 6 6 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 ζ3 ζ32 -1 -1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 1 1 1 ζ3 ζ32 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 -1 -1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 1 1 1 ζ32 ζ3 linear of order 6 ρ5 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ32 linear of order 3 ρ6 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ3 linear of order 3 ρ7 1 -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 linear of order 4 ρ8 1 -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 linear of order 4 ρ9 1 -1 1 1 ζ32 ζ3 -i i -1 -1 ζ6 ζ65 1 1 1 ζ32 ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 -1 -1 -1 ζ6 ζ65 linear of order 12 ρ10 1 -1 1 1 ζ32 ζ3 i -i -1 -1 ζ6 ζ65 1 1 1 ζ32 ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 -1 -1 -1 ζ6 ζ65 linear of order 12 ρ11 1 -1 1 1 ζ3 ζ32 i -i -1 -1 ζ65 ζ6 1 1 1 ζ3 ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 -1 -1 -1 ζ65 ζ6 linear of order 12 ρ12 1 -1 1 1 ζ3 ζ32 -i i -1 -1 ζ65 ζ6 1 1 1 ζ3 ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 -1 -1 -1 ζ65 ζ6 linear of order 12 ρ13 2 2 2 2 2 2 0 0 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 -2 2 2 2 2 0 0 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 2 -1+√-3 -1-√-3 0 0 2 2 -1+√-3 -1-√-3 -1 -1 -1 ζ65 ζ6 0 0 0 0 -1 -1 -1 ζ65 ζ6 complex lifted from C3×S3 ρ16 2 -2 2 2 -1-√-3 -1+√-3 0 0 -2 -2 1+√-3 1-√-3 -1 -1 -1 ζ6 ζ65 0 0 0 0 1 1 1 ζ32 ζ3 complex lifted from C3×Dic3 ρ17 2 -2 2 2 -1+√-3 -1-√-3 0 0 -2 -2 1-√-3 1+√-3 -1 -1 -1 ζ65 ζ6 0 0 0 0 1 1 1 ζ3 ζ32 complex lifted from C3×Dic3 ρ18 2 2 2 2 -1-√-3 -1+√-3 0 0 2 2 -1-√-3 -1+√-3 -1 -1 -1 ζ6 ζ65 0 0 0 0 -1 -1 -1 ζ6 ζ65 complex lifted from C3×S3 ρ19 6 6 -3 0 0 0 0 0 -3 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 orthogonal lifted from He3.S3 ρ20 6 6 6 -3 0 0 0 0 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ21 6 6 -3 0 0 0 0 0 -3 0 0 0 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 0 orthogonal lifted from He3.S3 ρ22 6 6 -3 0 0 0 0 0 -3 0 0 0 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 0 orthogonal lifted from He3.S3 ρ23 6 -6 6 -3 0 0 0 0 -6 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊C12, Schur index 2 ρ24 6 -6 -3 0 0 0 0 0 3 0 0 0 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 symplectic faithful, Schur index 2 ρ25 6 -6 -3 0 0 0 0 0 3 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 -3 0 0 0 0 0 3 0 0 0 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of He3.Dic3
On 108 points
Generators in S108
(1 70 33)(2 71 34)(3 72 35)(4 55 36)(5 56 19)(6 57 20)(7 58 21)(8 59 22)(9 60 23)(10 61 24)(11 62 25)(12 63 26)(13 64 27)(14 65 28)(15 66 29)(16 67 30)(17 68 31)(18 69 32)(37 91 79)(38 92 80)(39 93 81)(40 94 82)(41 95 83)(42 96 84)(43 97 85)(44 98 86)(45 99 87)(46 100 88)(47 101 89)(48 102 90)(49 103 73)(50 104 74)(51 105 75)(52 106 76)(53 107 77)(54 108 78)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)
(2 71 28)(3 35 60)(5 56 31)(6 20 63)(8 59 34)(9 23 66)(11 62 19)(12 26 69)(14 65 22)(15 29 72)(17 68 25)(18 32 57)(21 27 33)(24 30 36)(38 92 74)(39 81 99)(41 95 77)(42 84 102)(44 98 80)(45 87 105)(47 101 83)(48 90 108)(50 104 86)(51 75 93)(53 107 89)(54 78 96)(55 67 61)(58 70 64)(73 79 85)(76 82 88)(91 103 97)(94 106 100)
(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 52 10 43)(2 51 11 42)(3 50 12 41)(4 49 13 40)(5 48 14 39)(6 47 15 38)(7 46 16 37)(8 45 17 54)(9 44 18 53)(19 102 28 93)(20 101 29 92)(21 100 30 91)(22 99 31 108)(23 98 32 107)(24 97 33 106)(25 96 34 105)(26 95 35 104)(27 94 36 103)(55 73 64 82)(56 90 65 81)(57 89 66 80)(58 88 67 79)(59 87 68 78)(60 86 69 77)(61 85 70 76)(62 84 71 75)(63 83 72 74)

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

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

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

Matrix representation of He3.Dic3 in GL6(𝔽37)

 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 0 1 0 0 0 0 36 36 0 0 0 0 0 0 0 1 0 0 0 0 36 36 0 0 0 0 0 0 0 1 0 0 0 0 36 36
,
 1 0 0 0 0 0 0 1 0 0 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
,
 34 20 23 3 34 20 17 14 34 20 17 14 34 20 34 20 23 3 17 14 17 14 34 20 23 3 34 20 34 20 34 20 17 14 17 14
,
 0 31 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 31 0 0 0 0 31 0 0 0 0 31 0 0 0 0 31 0 0 0

G:=sub<GL(6,GF(37))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,36,0,0,0,0,1,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,1,36],[34,17,34,17,23,34,20,14,20,14,3,20,23,34,34,17,34,17,3,20,20,14,20,14,34,17,23,34,34,17,20,14,3,20,20,14],[0,31,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31,0,0,0] >;

He3.Dic3 in GAP, Magma, Sage, TeX

{\rm He}_3.{\rm Dic}_3
% in TeX

G:=Group("He3.Dic3");
// GroupNames label

G:=SmallGroup(324,16);
// by ID

G=gap.SmallGroup(324,16);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,5763,585,237,2164,2170,7781]);
// Polycyclic

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

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