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## G = C33.Dic3order 324 = 22·34

### 4th non-split extension by C33 of Dic3 acting via Dic3/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C33.Dic3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×3- 1+2 — C33.Dic3
 Lower central C3×C9 — C33.Dic3
 Upper central C1 — C2

Generators and relations for C33.Dic3
G = < a,b,c,d,e | a3=b3=c3=1, d6=c, e2=cd3, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=c-1d5 >

Subgroups: 252 in 78 conjugacy classes, 34 normal (16 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C9, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, Dic9, C3×Dic3, C3⋊Dic3, C3×C18, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C3×3- 1+2, C9⋊C12, C9⋊Dic3, C3×C3⋊Dic3, C6×3- 1+2, C33.Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, C9⋊C6, C3×C3⋊S3, C9⋊C12, C3×C3⋊Dic3, C33.S3, C33.Dic3

Smallest permutation representation of C33.Dic3
On 108 points
Generators in S108
(1 68 46)(2 57 41)(3 64 54)(4 71 49)(5 60 44)(6 67 39)(7 56 52)(8 63 47)(9 70 42)(10 59 37)(11 66 50)(12 55 45)(13 62 40)(14 69 53)(15 58 48)(16 65 43)(17 72 38)(18 61 51)(19 104 80)(20 93 75)(21 100 88)(22 107 83)(23 96 78)(24 103 73)(25 92 86)(26 99 81)(27 106 76)(28 95 89)(29 102 84)(30 91 79)(31 98 74)(32 105 87)(33 94 82)(34 101 77)(35 108 90)(36 97 85)
(1 46 68)(2 47 69)(3 48 70)(4 49 71)(5 50 72)(6 51 55)(7 52 56)(8 53 57)(9 54 58)(10 37 59)(11 38 60)(12 39 61)(13 40 62)(14 41 63)(15 42 64)(16 43 65)(17 44 66)(18 45 67)(19 98 86)(20 99 87)(21 100 88)(22 101 89)(23 102 90)(24 103 73)(25 104 74)(26 105 75)(27 106 76)(28 107 77)(29 108 78)(30 91 79)(31 92 80)(32 93 81)(33 94 82)(34 95 83)(35 96 84)(36 97 85)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(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 79 85)(74 80 86)(75 81 87)(76 82 88)(77 83 89)(78 84 90)(91 97 103)(92 98 104)(93 99 105)(94 100 106)(95 101 107)(96 102 108)
(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)
(1 106 10 97)(2 105 11 96)(3 104 12 95)(4 103 13 94)(5 102 14 93)(6 101 15 92)(7 100 16 91)(8 99 17 108)(9 98 18 107)(19 45 28 54)(20 44 29 53)(21 43 30 52)(22 42 31 51)(23 41 32 50)(24 40 33 49)(25 39 34 48)(26 38 35 47)(27 37 36 46)(55 89 64 80)(56 88 65 79)(57 87 66 78)(58 86 67 77)(59 85 68 76)(60 84 69 75)(61 83 70 74)(62 82 71 73)(63 81 72 90)

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

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

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

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 9A ··· 9I 12A 12B 12C 12D 18A ··· 18I order 1 2 3 3 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 size 1 1 2 2 2 2 3 3 6 6 27 27 2 2 2 2 3 3 6 6 6 ··· 6 27 27 27 27 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 type + + + + - - + - image C1 C2 C3 C4 C6 C12 S3 S3 Dic3 Dic3 C3×S3 C3×S3 C3×Dic3 C3×Dic3 C9⋊C6 C9⋊C12 kernel C33.Dic3 C6×3- 1+2 C9⋊Dic3 C3×3- 1+2 C3×C18 C3×C9 C2×3- 1+2 C32×C6 3- 1+2 C33 C18 C3×C6 C9 C32 C6 C3 # reps 1 1 2 2 2 4 3 1 3 1 6 2 6 2 3 3

Matrix representation of C33.Dic3 in GL10(𝔽37)

 10 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0
,
 11 0 0 0 0 0 0 0 0 0 12 27 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 32 18 0 0 0 0 0 0 0 0 15 5 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 1 36 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0

G:=sub<GL(10,GF(37))| [10,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0],[11,12,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0],[32,15,0,0,0,0,0,0,0,0,18,5,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

C33.Dic3 in GAP, Magma, Sage, TeX

C_3^3.{\rm Dic}_3
% in TeX

G:=Group("C3^3.Dic3");
// GroupNames label

G:=SmallGroup(324,100);
// by ID

G=gap.SmallGroup(324,100);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,3171,1449,453,2164,7781]);
// Polycyclic

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

׿
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