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## G = C3×C4⋊D12order 288 = 25·32

### Direct product of C3 and C4⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C4⋊D12
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C3×C4⋊D12
 Lower central C3 — C2×C6 — C3×C4⋊D12
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C4⋊D12
G = < a,b,c,d | a3=b4=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 690 in 231 conjugacy classes, 82 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×6], C22, C22 [×12], S3 [×4], C6 [×6], C6 [×7], C2×C4 [×3], D4 [×12], C23 [×4], C32, C12 [×12], C12 [×6], D6 [×12], C2×C6 [×2], C2×C6 [×13], C42, C2×D4 [×6], C3×S3 [×4], C3×C6 [×3], D12 [×12], C2×C12 [×6], C2×C12 [×3], C3×D4 [×12], C22×S3 [×4], C22×C6 [×4], C41D4, C3×C12 [×6], S3×C6 [×12], C62, C4×C12 [×2], C4×C12, C2×D12 [×6], C6×D4 [×6], C3×D12 [×12], C6×C12 [×3], S3×C2×C6 [×4], C4⋊D12, C3×C41D4, C122, C6×D12 [×6], C3×C4⋊D12
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×6], C23, D6 [×3], C2×C6 [×7], C2×D4 [×3], C3×S3, D12 [×6], C3×D4 [×6], C22×S3, C22×C6, C41D4, S3×C6 [×3], C2×D12 [×3], C6×D4 [×3], C3×D12 [×6], S3×C2×C6, C4⋊D12, C3×C41D4, C6×D12 [×3], C3×C4⋊D12

Smallest permutation representation of C3×C4⋊D12
On 96 points
Generators in S96
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(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 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 74 93 13)(2 75 94 14)(3 76 95 15)(4 77 96 16)(5 78 85 17)(6 79 86 18)(7 80 87 19)(8 81 88 20)(9 82 89 21)(10 83 90 22)(11 84 91 23)(12 73 92 24)(25 41 55 72)(26 42 56 61)(27 43 57 62)(28 44 58 63)(29 45 59 64)(30 46 60 65)(31 47 49 66)(32 48 50 67)(33 37 51 68)(34 38 52 69)(35 39 53 70)(36 40 54 71)
(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)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 36)(10 35)(11 34)(12 33)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(49 94)(50 93)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 96)(60 95)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 84)(70 83)(71 82)(72 81)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A ··· 4F 6A ··· 6F 6G ··· 6O 6P ··· 6W 12A ··· 12AV order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 12 12 12 12 1 1 2 2 2 2 ··· 2 1 ··· 1 2 ··· 2 12 ··· 12 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 D12 C3×D4 S3×C6 C3×D12 kernel C3×C4⋊D12 C122 C6×D12 C4⋊D12 C4×C12 C2×D12 C4×C12 C3×C12 C2×C12 C42 C12 C12 C2×C4 C4 # reps 1 1 6 2 2 12 1 6 3 2 12 12 6 24

Matrix representation of C3×C4⋊D12 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 5
,
 11 0 0 0 0 6 0 0 0 0 10 0 0 0 0 4
,
 0 6 0 0 11 0 0 0 0 0 0 4 0 0 10 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,5],[11,0,0,0,0,6,0,0,0,0,10,0,0,0,0,4],[0,11,0,0,6,0,0,0,0,0,0,10,0,0,4,0] >;

C3×C4⋊D12 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes D_{12}
% in TeX

G:=Group("C3xC4:D12");
// GroupNames label

G:=SmallGroup(288,645);
// by ID

G=gap.SmallGroup(288,645);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,142,9414]);
// Polycyclic

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

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