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## G = C3×C12⋊D4order 288 = 25·32

### Direct product of C3 and C12⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×C12⋊D4
 Lower central C3 — C2×C6 — C3×C12⋊D4
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×C12⋊D4
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd=b-1, dcd=c-1 >

Subgroups: 562 in 201 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22 [×10], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], C32, Dic3, C12 [×4], C12 [×9], D6 [×2], D6 [×8], C2×C6 [×2], C2×C6 [×11], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×4], C3×C6 [×3], C4×S3 [×2], D12 [×6], C2×Dic3, C2×C12 [×2], C2×C12 [×4], C2×C12 [×6], C3×D4 [×6], C22×S3, C22×S3 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3, C3×C12 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×8], C62, D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12 [×2], C22×C12, C6×D4 [×3], S3×C12 [×2], C3×D12 [×6], C6×Dic3, C6×C12, C6×C12 [×2], S3×C2×C6, S3×C2×C6 [×2], C12⋊D4, C3×C4⋊D4, C3×D6⋊C4 [×2], C32×C4⋊C4, S3×C2×C12, C6×D12, C6×D12 [×2], C3×C12⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×4], C23, D6 [×3], C2×C6 [×7], C2×D4 [×2], C4○D4, C3×S3, D12 [×2], C3×D4 [×4], C22×S3, C22×C6, C4⋊D4, S3×C6 [×3], C2×D12, S3×D4, Q83S3, C6×D4 [×2], C3×C4○D4, C3×D12 [×2], S3×C2×C6, C12⋊D4, C3×C4⋊D4, C6×D12, C3×S3×D4, C3×Q83S3, C3×C12⋊D4

Smallest permutation representation of C3×C12⋊D4
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(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 95 32 75)(2 90 33 82)(3 85 34 77)(4 92 35 84)(5 87 36 79)(6 94 25 74)(7 89 26 81)(8 96 27 76)(9 91 28 83)(10 86 29 78)(11 93 30 73)(12 88 31 80)(13 50 69 47)(14 57 70 42)(15 52 71 37)(16 59 72 44)(17 54 61 39)(18 49 62 46)(19 56 63 41)(20 51 64 48)(21 58 65 43)(22 53 66 38)(23 60 67 45)(24 55 68 40)
(1 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 24)(12 23)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 84)(57 83)(58 82)(59 81)(60 80)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 6T 6U 6V 6W 12A 12B 12C 12D 12E ··· 12Z 12AA 12AB 12AC 12AD order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 12 12 1 1 2 2 2 2 2 4 4 6 6 1 ··· 1 2 ··· 2 6 6 6 6 12 12 12 12 2 2 2 2 4 ··· 4 6 6 6 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D4 D6 C4○D4 C3×S3 D12 C3×D4 C3×D4 S3×C6 C3×C4○D4 C3×D12 S3×D4 Q8⋊3S3 C3×S3×D4 C3×Q8⋊3S3 kernel C3×C12⋊D4 C3×D6⋊C4 C32×C4⋊C4 S3×C2×C12 C6×D12 C12⋊D4 D6⋊C4 C3×C4⋊C4 S3×C2×C4 C2×D12 C3×C4⋊C4 C3×C12 S3×C6 C2×C12 C3×C6 C4⋊C4 C12 C12 D6 C2×C4 C6 C4 C6 C6 C2 C2 # reps 1 2 1 1 3 2 4 2 2 6 1 2 2 3 2 2 4 4 4 6 4 8 1 1 2 2

Matrix representation of C3×C12⋊D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 10 0 0 0 0 10 4 0 0 0 0 0 0 3 0 0 0 0 0 4 9 0 0 0 0 0 0 7 9 0 0 0 0 6 6
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 2 0 0 0 0 0 12
,
 2 6 0 0 0 0 6 11 0 0 0 0 0 0 12 5 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,10,0,0,0,0,10,4,0,0,0,0,0,0,3,4,0,0,0,0,0,9,0,0,0,0,0,0,7,6,0,0,0,0,9,6],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,12],[2,6,0,0,0,0,6,11,0,0,0,0,0,0,12,0,0,0,0,0,5,1,0,0,0,0,0,0,1,0,0,0,0,0,2,12] >;

C3×C12⋊D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes D_4
% in TeX

G:=Group("C3xC12:D4");
// GroupNames label

G:=SmallGroup(288,666);
// by ID

G=gap.SmallGroup(288,666);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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