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## G = C2×C6×Dic6order 288 = 25·32

### Direct product of C2×C6 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C6×Dic6
 Chief series C1 — C3 — C6 — C3×C6 — C3×Dic3 — C6×Dic3 — Dic3×C2×C6 — C2×C6×Dic6
 Lower central C3 — C6 — C2×C6×Dic6
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C2×C6×Dic6
G = < a,b,c,d | a2=b6=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 570 in 339 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×4], C4 [×8], C22 [×7], C6 [×2], C6 [×12], C6 [×7], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C32, Dic3 [×8], C12 [×8], C12 [×12], C2×C6 [×14], C2×C6 [×7], C22×C4, C22×C4 [×2], C2×Q8 [×12], C3×C6, C3×C6 [×6], Dic6 [×16], C2×Dic3 [×12], C2×C12 [×12], C2×C12 [×18], C3×Q8 [×16], C22×C6 [×2], C22×C6, C22×Q8, C3×Dic3 [×8], C3×C12 [×4], C62 [×7], C2×Dic6 [×12], C22×Dic3 [×2], C22×C12 [×2], C22×C12 [×3], C6×Q8 [×12], C3×Dic6 [×16], C6×Dic3 [×12], C6×C12 [×6], C2×C62, C22×Dic6, Q8×C2×C6, C6×Dic6 [×12], Dic3×C2×C6 [×2], C2×C6×C12, C2×C6×Dic6
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], Q8 [×4], C23 [×15], D6 [×7], C2×C6 [×35], C2×Q8 [×6], C24, C3×S3, Dic6 [×4], C3×Q8 [×4], C22×S3 [×7], C22×C6 [×15], C22×Q8, S3×C6 [×7], C2×Dic6 [×6], C6×Q8 [×6], S3×C23, C23×C6, C3×Dic6 [×4], S3×C2×C6 [×7], C22×Dic6, Q8×C2×C6, C6×Dic6 [×6], S3×C22×C6, C2×C6×Dic6

Smallest permutation representation of C2×C6×Dic6
On 96 points
Generators in S96
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 49)(9 50)(10 51)(11 52)(12 53)(13 80)(14 81)(15 82)(16 83)(17 84)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(1 29 9 25 5 33)(2 30 10 26 6 34)(3 31 11 27 7 35)(4 32 12 28 8 36)(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 60 45 56 41 52)(38 49 46 57 42 53)(39 50 47 58 43 54)(40 51 48 59 44 55)(61 76 65 80 69 84)(62 77 66 81 70 73)(63 78 67 82 71 74)(64 79 68 83 72 75)
(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 19 7 13)(2 18 8 24)(3 17 9 23)(4 16 10 22)(5 15 11 21)(6 14 12 20)(25 85 31 91)(26 96 32 90)(27 95 33 89)(28 94 34 88)(29 93 35 87)(30 92 36 86)(37 65 43 71)(38 64 44 70)(39 63 45 69)(40 62 46 68)(41 61 47 67)(42 72 48 66)(49 79 55 73)(50 78 56 84)(51 77 57 83)(52 76 58 82)(53 75 59 81)(54 74 60 80)

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

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

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

108 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4L 6A ··· 6N 6O ··· 6AI 12A ··· 12AF 12AG ··· 12AV order 1 2 ··· 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 2 2 2 2 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

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 Q8 D6 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C6 C3×Dic6 kernel C2×C6×Dic6 C6×Dic6 Dic3×C2×C6 C2×C6×C12 C22×Dic6 C2×Dic6 C22×Dic3 C22×C12 C22×C12 C62 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C4 C23 C22 # reps 1 12 2 1 2 24 4 2 1 4 6 1 2 8 8 12 2 16

Matrix representation of C2×C6×Dic6 in GL5(𝔽13)

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

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

C2×C6×Dic6 in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm Dic}_6
% in TeX

G:=Group("C2xC6xDic6");
// GroupNames label

G:=SmallGroup(288,988);
// by ID

G=gap.SmallGroup(288,988);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,336,1571,192,9414]);
// Polycyclic

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

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