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## G = C2×C6×D12order 288 = 25·32

### Direct product of C2×C6 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C6×D12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C22×C6 — C2×C6×D12
 Lower central C3 — C6 — C2×C6×D12
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C2×C6×D12
G = < a,b,c,d | a2=b6=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1210 in 499 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×2], C3, C4 [×4], C22 [×7], C22 [×32], S3 [×8], C6 [×2], C6 [×12], C6 [×15], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, C12 [×8], C12 [×4], D6 [×8], D6 [×24], C2×C6 [×14], C2×C6 [×39], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×8], C3×C6, C3×C6 [×6], D12 [×16], C2×C12 [×12], C2×C12 [×6], C3×D4 [×16], C22×S3 [×12], C22×S3 [×8], C22×C6 [×2], C22×C6 [×21], C22×D4, C3×C12 [×4], S3×C6 [×8], S3×C6 [×24], C62 [×7], C2×D12 [×12], C22×C12 [×2], C22×C12, C6×D4 [×12], S3×C23 [×2], C23×C6 [×2], C3×D12 [×16], C6×C12 [×6], S3×C2×C6 [×12], S3×C2×C6 [×8], C2×C62, C22×D12, D4×C2×C6, C6×D12 [×12], C2×C6×C12, S3×C22×C6 [×2], C2×C6×D12
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], D4 [×4], C23 [×15], D6 [×7], C2×C6 [×35], C2×D4 [×6], C24, C3×S3, D12 [×4], C3×D4 [×4], C22×S3 [×7], C22×C6 [×15], C22×D4, S3×C6 [×7], C2×D12 [×6], C6×D4 [×6], S3×C23, C23×C6, C3×D12 [×4], S3×C2×C6 [×7], C22×D12, D4×C2×C6, C6×D12 [×6], S3×C22×C6, C2×C6×D12

Smallest permutation representation of C2×C6×D12
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 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 85)(36 86)(37 80)(38 81)(39 82)(40 83)(41 84)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 68)(50 69)(51 70)(52 71)(53 72)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(1 31 5 35 9 27)(2 32 6 36 10 28)(3 33 7 25 11 29)(4 34 8 26 12 30)(13 87 17 91 21 95)(14 88 18 92 22 96)(15 89 19 93 23 85)(16 90 20 94 24 86)(37 56 45 52 41 60)(38 57 46 53 42 49)(39 58 47 54 43 50)(40 59 48 55 44 51)(61 74 69 82 65 78)(62 75 70 83 66 79)(63 76 71 84 67 80)(64 77 72 73 68 81)
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 48)(10 47)(11 46)(12 45)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 84)(23 83)(24 82)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 60)(35 59)(36 58)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)

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

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

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

108 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6N 6O ··· 6AI 6AJ ··· 6AY 12A ··· 12AF order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 ··· 6 1 1 2 2 2 2 2 2 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D6 C3×S3 D12 C3×D4 S3×C6 S3×C6 C3×D12 kernel C2×C6×D12 C6×D12 C2×C6×C12 S3×C22×C6 C22×D12 C2×D12 C22×C12 S3×C23 C22×C12 C62 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C4 C23 C22 # reps 1 12 1 2 2 24 2 4 1 4 6 1 2 8 8 12 2 16

Matrix representation of C2×C6×D12 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 10 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 9 0 0 0 0 0 9
,
 12 0 0 0 0 0 4 0 0 0 0 3 10 0 0 0 0 0 2 0 0 0 0 8 7
,
 1 0 0 0 0 0 3 6 0 0 0 3 10 0 0 0 0 0 9 4 0 0 0 6 4

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[10,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9],[12,0,0,0,0,0,4,3,0,0,0,0,10,0,0,0,0,0,2,8,0,0,0,0,7],[1,0,0,0,0,0,3,3,0,0,0,6,10,0,0,0,0,0,9,6,0,0,0,4,4] >;

C2×C6×D12 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_{12}
% in TeX

G:=Group("C2xC6xD12");
// GroupNames label

G:=SmallGroup(288,990);
// by ID

G=gap.SmallGroup(288,990);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,1571,192,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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