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## G = C2×Dic3⋊Dic3order 288 = 25·32

### Direct product of C2 and Dic3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×Dic3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C2×Dic3⋊Dic3
 Lower central C32 — C3×C6 — C2×Dic3⋊Dic3
 Upper central C1 — C23

Generators and relations for C2×Dic3⋊Dic3
G = < a,b,c,d,e | a2=b6=d6=1, c2=b3, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >

Subgroups: 594 in 211 conjugacy classes, 100 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×8], C22, C22 [×6], C6 [×6], C6 [×8], C6 [×7], C2×C4 [×14], C23, C32, Dic3 [×4], Dic3 [×10], C12 [×6], C2×C6 [×2], C2×C6 [×12], C2×C6 [×7], C4⋊C4 [×4], C22×C4 [×3], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×8], C2×Dic3 [×16], C2×C12 [×10], C22×C6 [×2], C22×C6, C2×C4⋊C4, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3 [×2], C62, C62 [×6], Dic3⋊C4 [×4], C4⋊Dic3 [×4], C22×Dic3 [×2], C22×Dic3 [×3], C22×C12 [×2], C6×Dic3 [×8], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C62, C2×Dic3⋊C4, C2×C4⋊Dic3, Dic3⋊Dic3 [×4], Dic3×C2×C6 [×2], C22×C3⋊Dic3, C2×Dic3⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, Dic3 [×4], D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×4], C4×S3 [×2], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×2], C22×S3 [×2], C2×C4⋊C4, S32, Dic3⋊C4 [×4], C4⋊Dic3 [×4], C2×Dic6 [×2], S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, S3×Dic3 [×2], C3⋊D12 [×2], C322Q8 [×2], C2×S32, C2×Dic3⋊C4, C2×C4⋊Dic3, Dic3⋊Dic3 [×4], C2×S3×Dic3, C2×C3⋊D12, C2×C322Q8, C2×Dic3⋊Dic3

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

60 irreducible representations

 dim 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 C4 S3 D4 Q8 Dic3 D6 D6 Dic6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 C3⋊D12 C32⋊2Q8 C2×S32 kernel C2×Dic3⋊Dic3 Dic3⋊Dic3 Dic3×C2×C6 C22×C3⋊Dic3 C6×Dic3 C22×Dic3 C62 C62 C2×Dic3 C2×Dic3 C22×C6 C2×C6 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C22 # reps 1 4 2 1 8 2 2 2 4 4 2 8 4 4 4 1 2 2 2 1

Matrix representation of C2×Dic3⋊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 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 9 10 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 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 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
,
 0 1 0 0 0 0 0 0 12 0 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 1 0 0 0 0 0 0 0 12 12

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],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,10,0,0,0,0,0,0,10,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;

C2×Dic3⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic3:Dic3");
// GroupNames label

G:=SmallGroup(288,613);
// by ID

G=gap.SmallGroup(288,613);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^6=1,c^2=b^3,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=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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