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## G = C2×D6.Dic3order 288 = 25·32

### Direct product of C2 and D6.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×D6.Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.Dic3 — C2×D6.Dic3
 Lower central C32 — C3×C6 — C2×D6.Dic3
 Upper central C1 — C2×C4

Generators and relations for C2×D6.Dic3
G = < a,b,c,d,e | a2=b6=c2=1, d6=b3, e2=b3d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d5 >

Subgroups: 370 in 147 conjugacy classes, 68 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C2×C8 [×2], M4(2) [×4], C22×C4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C3⋊C8 [×6], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C8⋊S3 [×4], C2×C3⋊C8, C2×C3⋊C8 [×3], C4.Dic3 [×4], C2×C24, S3×C2×C4, C22×C12, C3×C3⋊C8 [×2], C324C8 [×2], S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, C2×C8⋊S3, C2×C4.Dic3, D6.Dic3 [×4], C6×C3⋊C8, C2×C324C8, S3×C2×C12, C2×D6.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], M4(2) [×2], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C2×M4(2), S32, C8⋊S3 [×2], C4.Dic3 [×2], S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, C2×C8⋊S3, C2×C4.Dic3, D6.Dic3 [×2], C2×S3×Dic3, C2×D6.Dic3

Smallest permutation representation of C2×D6.Dic3
On 96 points
Generators in S96
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 85)(81 86)(82 87)(83 88)(84 89)
(1 3 5 7 9 11)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 16 18 20 22 24)(25 35 33 31 29 27)(26 36 34 32 30 28)(37 47 45 43 41 39)(38 48 46 44 42 40)(49 59 57 55 53 51)(50 60 58 56 54 52)(61 71 69 67 65 63)(62 72 70 68 66 64)(73 75 77 79 81 83)(74 76 78 80 82 84)(85 87 89 91 93 95)(86 88 90 92 94 96)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 61)(9 62)(10 63)(11 64)(12 65)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 49)(21 50)(22 51)(23 52)(24 53)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 85)(35 86)(36 87)(37 80)(38 81)(39 82)(40 83)(41 84)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)
(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)
(1 32 10 29 7 26 4 35)(2 25 11 34 8 31 5 28)(3 30 12 27 9 36 6 33)(13 41 22 38 19 47 16 44)(14 46 23 43 20 40 17 37)(15 39 24 48 21 45 18 42)(49 77 58 74 55 83 52 80)(50 82 59 79 56 76 53 73)(51 75 60 84 57 81 54 78)(61 88 70 85 67 94 64 91)(62 93 71 90 68 87 65 96)(63 86 72 95 69 92 66 89)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I 12J 12K 12L 12M 12N 12O 12P 24A ··· 24H order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 6 6 2 2 4 1 1 1 1 6 6 2 ··· 2 4 4 4 6 6 6 6 6 6 6 6 18 18 18 18 2 ··· 2 4 4 4 4 6 6 6 6 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + - + - + - + - image C1 C2 C2 C2 C2 C4 C4 C4 S3 S3 D6 Dic3 D6 Dic3 D6 Dic3 M4(2) C4×S3 C4×S3 C8⋊S3 C4.Dic3 S32 S3×Dic3 C2×S32 S3×Dic3 D6.Dic3 kernel C2×D6.Dic3 D6.Dic3 C6×C3⋊C8 C2×C32⋊4C8 S3×C2×C12 S3×C12 C6×Dic3 S3×C2×C6 C2×C3⋊C8 S3×C2×C4 C3⋊C8 C4×S3 C4×S3 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 C2×C6 C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 1 1 2 2 2 1 2 1 4 2 2 8 8 1 1 1 1 4

Matrix representation of C2×D6.Dic3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 46 47 0 0 0 0 28 27 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 1 0 0 0 0 46 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,28,0,0,0,0,47,27,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,46,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D6.Dic3 in GAP, Magma, Sage, TeX

C_2\times D_6.{\rm Dic}_3
% in TeX

G:=Group("C2xD6.Dic3");
// GroupNames label

G:=SmallGroup(288,467);
// by ID

G=gap.SmallGroup(288,467);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,80,1356,9414]);
// Polycyclic

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

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