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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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C8⋊S3, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C2×C24, S3×C2×C4, C22×C12, C3×C3⋊C8, C324C8, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, C2×C8⋊S3, C2×C4.Dic3, D6.Dic3, C6×C3⋊C8, C2×C324C8, S3×C2×C12, C2×D6.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, C4×S3, C2×Dic3, C22×S3, C2×M4(2), S32, C8⋊S3, C4.Dic3, S3×C2×C4, C22×Dic3, S3×Dic3, C2×S32, C2×C8⋊S3, C2×C4.Dic3, D6.Dic3, C2×S3×Dic3, C2×D6.Dic3

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

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

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

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

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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