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## G = C6×D24order 288 = 25·32

### Direct product of C6 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C6×D24
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C6×D12 — C6×D24
 Lower central C3 — C6 — C12 — C6×D24
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C6×D24
G = < a,b,c | a6=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 538 in 163 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6 [×2], C6 [×4], C6 [×7], C8 [×2], C2×C4, D4 [×6], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×8], C2×C6 [×2], C2×C6 [×9], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×4], C3×C6, C3×C6 [×2], C24 [×4], C24 [×2], D12 [×4], D12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×6], C22×S3 [×2], C22×C6 [×2], C2×D8, C3×C12 [×2], S3×C6 [×8], C62, D24 [×4], C2×C24 [×2], C2×C24, C3×D8 [×4], C2×D12 [×2], C6×D4 [×2], C3×C24 [×2], C3×D12 [×4], C3×D12 [×2], C6×C12, S3×C2×C6 [×2], C2×D24, C6×D8, C3×D24 [×4], C6×C24, C6×D12 [×2], C6×D24
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], D8 [×2], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×D8, S3×C6 [×3], D24 [×2], C3×D8 [×2], C2×D12, C6×D4, C3×D12 [×2], S3×C2×C6, C2×D24, C6×D8, C3×D24 [×2], C6×D12, C6×D24

Smallest permutation representation of C6×D24
On 96 points
Generators in S96
(1 96 9 80 17 88)(2 73 10 81 18 89)(3 74 11 82 19 90)(4 75 12 83 20 91)(5 76 13 84 21 92)(6 77 14 85 22 93)(7 78 15 86 23 94)(8 79 16 87 24 95)(25 70 41 62 33 54)(26 71 42 63 34 55)(27 72 43 64 35 56)(28 49 44 65 36 57)(29 50 45 66 37 58)(30 51 46 67 38 59)(31 52 47 68 39 60)(32 53 48 69 40 61)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 80)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6O 6P ··· 6W 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 12 12 12 12 1 1 2 2 2 2 2 1 ··· 1 2 ··· 2 12 ··· 12 2 2 2 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D4 D6 D6 D8 C3×S3 D12 C3×D4 D12 C3×D4 S3×C6 S3×C6 D24 C3×D8 C3×D12 C3×D12 C3×D24 kernel C6×D24 C3×D24 C6×C24 C6×D12 C2×D24 D24 C2×C24 C2×D12 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C2×C8 C12 C12 C2×C6 C2×C6 C8 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 1 1 1 2 1 4 2 2 2 2 2 4 2 8 8 4 4 16

Matrix representation of C6×D24 in GL5(𝔽73)

 9 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 24 0 0 0 0 0 70 0 0 0 0 0 0 41 0 0 0 16 41
,
 72 0 0 0 0 0 0 8 0 0 0 64 0 0 0 0 0 0 10 52 0 0 0 36 63

G:=sub<GL(5,GF(73))| [9,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,24,0,0,0,0,0,70,0,0,0,0,0,0,16,0,0,0,41,41],[72,0,0,0,0,0,0,64,0,0,0,8,0,0,0,0,0,0,10,36,0,0,0,52,63] >;

C6×D24 in GAP, Magma, Sage, TeX

C_6\times D_{24}
% in TeX

G:=Group("C6xD24");
// GroupNames label

G:=SmallGroup(288,674);
// by ID

G=gap.SmallGroup(288,674);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,394,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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