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## G = C6×D6⋊C4order 288 = 25·32

### Direct product of C6 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C22×C6 — C6×D6⋊C4
 Lower central C3 — C6 — C6×D6⋊C4
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C6×D6⋊C4
G = < a,b,c,d | a6=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 698 in 291 conjugacy classes, 114 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3 [×2], C3, C4 [×4], C22, C22 [×6], C22 [×16], S3 [×4], C6 [×6], C6 [×8], C6 [×11], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C32, Dic3 [×2], C12 [×10], D6 [×4], D6 [×12], C2×C6 [×2], C2×C6 [×12], C2×C6 [×23], C22⋊C4 [×4], C22×C4, C22×C4, C24, C3×S3 [×4], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×4], C2×C12 [×14], C22×S3 [×6], C22×S3 [×4], C22×C6 [×2], C22×C6 [×11], C2×C22⋊C4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×4], S3×C6 [×12], C62, C62 [×6], D6⋊C4 [×4], C3×C22⋊C4 [×4], C22×Dic3, C22×C12 [×2], C22×C12 [×2], S3×C23, C23×C6, C6×Dic3 [×2], C6×Dic3 [×2], C6×C12 [×2], C6×C12 [×2], S3×C2×C6 [×6], S3×C2×C6 [×4], C2×C62, C2×D6⋊C4, C6×C22⋊C4, C3×D6⋊C4 [×4], Dic3×C2×C6, C2×C6×C12, S3×C22×C6, C6×D6⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×4], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C3×S3, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12 [×6], C3×D4 [×4], C22×S3, C22×C6, C2×C22⋊C4, S3×C6 [×3], D6⋊C4 [×4], C3×C22⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4 [×2], S3×C12 [×2], C3×D12 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C2×D6⋊C4, C6×C22⋊C4, C3×D6⋊C4 [×4], S3×C2×C12, C6×D12, C6×C3⋊D4, C6×D6⋊C4

Smallest permutation representation of C6×D6⋊C4
On 96 points
Generators in S96
(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 9 3 11 5 7)(2 10 4 12 6 8)(13 50 15 52 17 54)(14 51 16 53 18 49)(19 35 21 31 23 33)(20 36 22 32 24 34)(25 62 29 66 27 64)(26 63 30 61 28 65)(37 45 39 47 41 43)(38 46 40 48 42 44)(55 72 59 70 57 68)(56 67 60 71 58 69)(73 90 77 88 75 86)(74 85 78 89 76 87)(79 96 83 94 81 92)(80 91 84 95 82 93)
(1 78)(2 73)(3 74)(4 75)(5 76)(6 77)(7 89)(8 90)(9 85)(10 86)(11 87)(12 88)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 56)(20 57)(21 58)(22 59)(23 60)(24 55)(31 71)(32 72)(33 67)(34 68)(35 69)(36 70)(37 93)(38 94)(39 95)(40 96)(41 91)(42 92)(43 80)(44 81)(45 82)(46 83)(47 84)(48 79)(49 62)(50 63)(51 64)(52 65)(53 66)(54 61)
(1 13 19 40)(2 14 20 41)(3 15 21 42)(4 16 22 37)(5 17 23 38)(6 18 24 39)(7 54 33 46)(8 49 34 47)(9 50 35 48)(10 51 36 43)(11 52 31 44)(12 53 32 45)(25 72 91 88)(26 67 92 89)(27 68 93 90)(28 69 94 85)(29 70 95 86)(30 71 96 87)(55 80 77 64)(56 81 78 65)(57 82 73 66)(58 83 74 61)(59 84 75 62)(60 79 76 63)

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

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

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

108 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6N 6O ··· 6AI 6AJ ··· 6AQ 12A ··· 12AF 12AG ··· 12AN order 1 2 ··· 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 6 6 6 6 1 1 2 2 2 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 D4 D6 D6 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 kernel C6×D6⋊C4 C3×D6⋊C4 Dic3×C2×C6 C2×C6×C12 S3×C22×C6 C2×D6⋊C4 S3×C2×C6 D6⋊C4 C22×Dic3 C22×C12 S3×C23 C22×S3 C22×C12 C62 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 2 8 8 2 2 2 16 1 4 2 1 2 4 4 4 8 4 2 8 8 8

Matrix representation of C6×D6⋊C4 in GL4(𝔽13) generated by

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

C6×D6⋊C4 in GAP, Magma, Sage, TeX

C_6\times D_6\rtimes C_4
% in TeX

G:=Group("C6xD6:C4");
// GroupNames label

G:=SmallGroup(288,698);
// by ID

G=gap.SmallGroup(288,698);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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