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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 — C62 — S3×C2×C6 — D6⋊Dic3 — C2×D6⋊Dic3
 Lower central C32 — C3×C6 — C2×D6⋊Dic3
 Upper central C1 — C23

Generators and relations for C2×D6⋊Dic3
G = < a,b,c,d,e | a2=b6=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >

Subgroups: 914 in 291 conjugacy classes, 100 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3 [×2], C3, C4 [×4], C22, C22 [×6], C22 [×16], S3 [×4], C6 [×6], C6 [×8], C6 [×11], C2×C4 [×8], C23, C23 [×10], C32, Dic3 [×10], C12 [×2], D6 [×4], D6 [×12], C2×C6 [×2], C2×C6 [×12], C2×C6 [×23], C22⋊C4 [×4], C22×C4 [×2], C24, C3×S3 [×4], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×2], C2×Dic3 [×16], C2×C12 [×4], C22×S3 [×6], C22×S3 [×4], C22×C6 [×2], C22×C6 [×11], C2×C22⋊C4, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×4], S3×C6 [×12], C62, C62 [×6], D6⋊C4 [×4], C6.D4 [×4], C22×Dic3, C22×Dic3 [×3], C22×C12, S3×C23, C23×C6, C6×Dic3 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6 [×6], S3×C2×C6 [×4], C2×C62, C2×D6⋊C4, C2×C6.D4, D6⋊Dic3 [×4], Dic3×C2×C6, C22×C3⋊Dic3, S3×C22×C6, C2×D6⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×6], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×6], C22×S3 [×2], C2×C22⋊C4, S32, D6⋊C4 [×4], C6.D4 [×4], S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4 [×3], S3×Dic3 [×2], D6⋊S3 [×2], C3⋊D12 [×2], C2×S32, C2×D6⋊C4, C2×C6.D4, D6⋊Dic3 [×4], C2×S3×Dic3, C2×D6⋊S3, C2×C3⋊D12, C2×D6⋊Dic3

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

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6N 6O ··· 6U 6V ··· 6AC 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 2 2 4 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + - - + + image C1 C2 C2 C2 C2 C4 S3 S3 D4 D6 Dic3 D6 D6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 D6⋊S3 C3⋊D12 C2×S32 kernel C2×D6⋊Dic3 D6⋊Dic3 Dic3×C2×C6 C22×C3⋊Dic3 S3×C22×C6 S3×C2×C6 C22×Dic3 S3×C23 C62 C2×Dic3 C22×S3 C22×S3 C22×C6 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C22 # reps 1 4 1 1 1 8 1 1 4 2 4 2 2 4 4 12 1 2 2 2 1

Matrix representation of C2×D6⋊Dic3 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 3 4 0 0 0 0 0 0 11 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[3,11,0,0,0,0,0,0,4,10,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C2×D6⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,608);
// by ID

G=gap.SmallGroup(288,608);
# by ID

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

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

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