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## G = C3×D6⋊C8order 288 = 25·32

### Direct product of C3 and D6⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D6⋊C8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — S3×C2×C12 — C3×D6⋊C8
 Lower central C3 — C6 — C3×D6⋊C8
 Upper central C1 — C2×C12 — C2×C24

Generators and relations for C3×D6⋊C8
G = < a,b,c,d | a3=b6=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 250 in 111 conjugacy classes, 50 normal (46 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C8 [×2], C2×C4, C2×C4 [×3], C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C2×C8, C2×C8, C22×C4, C3×S3 [×2], C3×C6 [×3], C3⋊C8, C24 [×5], C4×S3 [×2], C2×Dic3, C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C22⋊C8, C3×Dic3, C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C2×C3⋊C8, C2×C24 [×2], C2×C24 [×2], S3×C2×C4, C22×C12, C3×C3⋊C8, C3×C24, S3×C12 [×2], C6×Dic3, C6×C12, S3×C2×C6, D6⋊C8, C3×C22⋊C8, C6×C3⋊C8, C6×C24, S3×C2×C12, C3×D6⋊C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C2×C8, M4(2), C3×S3, C24 [×2], C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C22⋊C8, S3×C6, S3×C8, C8⋊S3, D6⋊C4, C3×C22⋊C4, C2×C24, C3×M4(2), S3×C12, C3×D12, C3×C3⋊D4, D6⋊C8, C3×C22⋊C8, S3×C24, C3×C8⋊S3, C3×D6⋊C4, C3×D6⋊C8

Smallest permutation representation of C3×D6⋊C8
On 96 points
Generators in S96
(1 62 43)(2 63 44)(3 64 45)(4 57 46)(5 58 47)(6 59 48)(7 60 41)(8 61 42)(9 86 96)(10 87 89)(11 88 90)(12 81 91)(13 82 92)(14 83 93)(15 84 94)(16 85 95)(17 79 55)(18 80 56)(19 73 49)(20 74 50)(21 75 51)(22 76 52)(23 77 53)(24 78 54)(25 35 68)(26 36 69)(27 37 70)(28 38 71)(29 39 72)(30 40 65)(31 33 66)(32 34 67)
(1 53 62 23 43 77)(2 54 63 24 44 78)(3 55 64 17 45 79)(4 56 57 18 46 80)(5 49 58 19 47 73)(6 50 59 20 48 74)(7 51 60 21 41 75)(8 52 61 22 42 76)(9 31 96 66 86 33)(10 32 89 67 87 34)(11 25 90 68 88 35)(12 26 91 69 81 36)(13 27 92 70 82 37)(14 28 93 71 83 38)(15 29 94 72 84 39)(16 30 95 65 85 40)
(1 90)(2 36)(3 92)(4 38)(5 94)(6 40)(7 96)(8 34)(9 60)(10 76)(11 62)(12 78)(13 64)(14 80)(15 58)(16 74)(17 37)(18 93)(19 39)(20 95)(21 33)(22 89)(23 35)(24 91)(25 53)(26 44)(27 55)(28 46)(29 49)(30 48)(31 51)(32 42)(41 86)(43 88)(45 82)(47 84)(50 85)(52 87)(54 81)(56 83)(57 71)(59 65)(61 67)(63 69)(66 75)(68 77)(70 79)(72 73)
(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)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I ··· 12T 12U 12V 12W 12X 24A ··· 24AF 24AG ··· 24AN order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 ··· 24 24 ··· 24 size 1 1 1 1 6 6 1 1 2 2 2 1 1 1 1 6 6 1 ··· 1 2 ··· 2 6 6 6 6 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 6 6 6 6 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 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 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 S3 D4 D6 M4(2) C3×S3 D12 C3⋊D4 C3×D4 C4×S3 S3×C6 S3×C8 C8⋊S3 C3×M4(2) C3×D12 C3×C3⋊D4 S3×C12 S3×C24 C3×C8⋊S3 kernel C3×D6⋊C8 C6×C3⋊C8 C6×C24 S3×C2×C12 D6⋊C8 C6×Dic3 S3×C2×C6 C2×C3⋊C8 C2×C24 S3×C2×C4 S3×C6 C2×Dic3 C22×S3 D6 C2×C24 C3×C12 C2×C12 C3×C6 C2×C8 C12 C12 C12 C2×C6 C2×C4 C6 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 2 2 2 2 2 8 4 4 16 1 2 1 2 2 2 2 4 2 2 4 4 4 4 4 4 8 8

Matrix representation of C3×D6⋊C8 in GL3(𝔽73) generated by

 8 0 0 0 8 0 0 0 8
,
 1 0 0 0 9 0 0 0 65
,
 72 0 0 0 0 65 0 9 0
,
 10 0 0 0 1 0 0 0 72
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,9,0,0,0,65],[72,0,0,0,0,9,0,65,0],[10,0,0,0,1,0,0,0,72] >;

C3×D6⋊C8 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes C_8
% in TeX

G:=Group("C3xD6:C8");
// GroupNames label

G:=SmallGroup(288,254);
// by ID

G=gap.SmallGroup(288,254);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,136,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^2=d^8=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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𝔽