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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 — C12 — C24 — C3×C24 — S3×C24 — C3×D6.C8
 Lower central C3 — C6 — C3×D6.C8
 Upper central C1 — C24 — C48

Generators and relations for C3×D6.C8
G = < a,b,c,d | a3=b6=c2=1, d8=b3, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 102 in 57 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C16, C16, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, M5(2), C3×Dic3, C3×C12, S3×C6, C3⋊C16, C48, C48, S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, D6.C8, C3×M5(2), C3×C3⋊C16, C3×C48, S3×C24, C3×D6.C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C24, C4×S3, C2×C12, M5(2), S3×C6, S3×C8, C2×C24, S3×C12, D6.C8, C3×M5(2), S3×C24, C3×D6.C8

Smallest permutation representation of C3×D6.C8
On 96 points
Generators in S96
(1 64 68)(2 49 69)(3 50 70)(4 51 71)(5 52 72)(6 53 73)(7 54 74)(8 55 75)(9 56 76)(10 57 77)(11 58 78)(12 59 79)(13 60 80)(14 61 65)(15 62 66)(16 63 67)(17 92 38)(18 93 39)(19 94 40)(20 95 41)(21 96 42)(22 81 43)(23 82 44)(24 83 45)(25 84 46)(26 85 47)(27 86 48)(28 87 33)(29 88 34)(30 89 35)(31 90 36)(32 91 37)
(1 76 64 9 68 56)(2 77 49 10 69 57)(3 78 50 11 70 58)(4 79 51 12 71 59)(5 80 52 13 72 60)(6 65 53 14 73 61)(7 66 54 15 74 62)(8 67 55 16 75 63)(17 84 38 25 92 46)(18 85 39 26 93 47)(19 86 40 27 94 48)(20 87 41 28 95 33)(21 88 42 29 96 34)(22 89 43 30 81 35)(23 90 44 31 82 36)(24 91 45 32 83 37)
(1 86)(2 95)(3 88)(4 81)(5 90)(6 83)(7 92)(8 85)(9 94)(10 87)(11 96)(12 89)(13 82)(14 91)(15 84)(16 93)(17 74)(18 67)(19 76)(20 69)(21 78)(22 71)(23 80)(24 73)(25 66)(26 75)(27 68)(28 77)(29 70)(30 79)(31 72)(32 65)(33 57)(34 50)(35 59)(36 52)(37 61)(38 54)(39 63)(40 56)(41 49)(42 58)(43 51)(44 60)(45 53)(46 62)(47 55)(48 64)
(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)| (1,64,68)(2,49,69)(3,50,70)(4,51,71)(5,52,72)(6,53,73)(7,54,74)(8,55,75)(9,56,76)(10,57,77)(11,58,78)(12,59,79)(13,60,80)(14,61,65)(15,62,66)(16,63,67)(17,92,38)(18,93,39)(19,94,40)(20,95,41)(21,96,42)(22,81,43)(23,82,44)(24,83,45)(25,84,46)(26,85,47)(27,86,48)(28,87,33)(29,88,34)(30,89,35)(31,90,36)(32,91,37), (1,76,64,9,68,56)(2,77,49,10,69,57)(3,78,50,11,70,58)(4,79,51,12,71,59)(5,80,52,13,72,60)(6,65,53,14,73,61)(7,66,54,15,74,62)(8,67,55,16,75,63)(17,84,38,25,92,46)(18,85,39,26,93,47)(19,86,40,27,94,48)(20,87,41,28,95,33)(21,88,42,29,96,34)(22,89,43,30,81,35)(23,90,44,31,82,36)(24,91,45,32,83,37), (1,86)(2,95)(3,88)(4,81)(5,90)(6,83)(7,92)(8,85)(9,94)(10,87)(11,96)(12,89)(13,82)(14,91)(15,84)(16,93)(17,74)(18,67)(19,76)(20,69)(21,78)(22,71)(23,80)(24,73)(25,66)(26,75)(27,68)(28,77)(29,70)(30,79)(31,72)(32,65)(33,57)(34,50)(35,59)(36,52)(37,61)(38,54)(39,63)(40,56)(41,49)(42,58)(43,51)(44,60)(45,53)(46,62)(47,55)(48,64), (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( (1,64,68)(2,49,69)(3,50,70)(4,51,71)(5,52,72)(6,53,73)(7,54,74)(8,55,75)(9,56,76)(10,57,77)(11,58,78)(12,59,79)(13,60,80)(14,61,65)(15,62,66)(16,63,67)(17,92,38)(18,93,39)(19,94,40)(20,95,41)(21,96,42)(22,81,43)(23,82,44)(24,83,45)(25,84,46)(26,85,47)(27,86,48)(28,87,33)(29,88,34)(30,89,35)(31,90,36)(32,91,37), (1,76,64,9,68,56)(2,77,49,10,69,57)(3,78,50,11,70,58)(4,79,51,12,71,59)(5,80,52,13,72,60)(6,65,53,14,73,61)(7,66,54,15,74,62)(8,67,55,16,75,63)(17,84,38,25,92,46)(18,85,39,26,93,47)(19,86,40,27,94,48)(20,87,41,28,95,33)(21,88,42,29,96,34)(22,89,43,30,81,35)(23,90,44,31,82,36)(24,91,45,32,83,37), (1,86)(2,95)(3,88)(4,81)(5,90)(6,83)(7,92)(8,85)(9,94)(10,87)(11,96)(12,89)(13,82)(14,91)(15,84)(16,93)(17,74)(18,67)(19,76)(20,69)(21,78)(22,71)(23,80)(24,73)(25,66)(26,75)(27,68)(28,77)(29,70)(30,79)(31,72)(32,65)(33,57)(34,50)(35,59)(36,52)(37,61)(38,54)(39,63)(40,56)(41,49)(42,58)(43,51)(44,60)(45,53)(46,62)(47,55)(48,64), (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([[(1,64,68),(2,49,69),(3,50,70),(4,51,71),(5,52,72),(6,53,73),(7,54,74),(8,55,75),(9,56,76),(10,57,77),(11,58,78),(12,59,79),(13,60,80),(14,61,65),(15,62,66),(16,63,67),(17,92,38),(18,93,39),(19,94,40),(20,95,41),(21,96,42),(22,81,43),(23,82,44),(24,83,45),(25,84,46),(26,85,47),(27,86,48),(28,87,33),(29,88,34),(30,89,35),(31,90,36),(32,91,37)], [(1,76,64,9,68,56),(2,77,49,10,69,57),(3,78,50,11,70,58),(4,79,51,12,71,59),(5,80,52,13,72,60),(6,65,53,14,73,61),(7,66,54,15,74,62),(8,67,55,16,75,63),(17,84,38,25,92,46),(18,85,39,26,93,47),(19,86,40,27,94,48),(20,87,41,28,95,33),(21,88,42,29,96,34),(22,89,43,30,81,35),(23,90,44,31,82,36),(24,91,45,32,83,37)], [(1,86),(2,95),(3,88),(4,81),(5,90),(6,83),(7,92),(8,85),(9,94),(10,87),(11,96),(12,89),(13,82),(14,91),(15,84),(16,93),(17,74),(18,67),(19,76),(20,69),(21,78),(22,71),(23,80),(24,73),(25,66),(26,75),(27,68),(28,77),(29,70),(30,79),(31,72),(32,65),(33,57),(34,50),(35,59),(36,52),(37,61),(38,54),(39,63),(40,56),(41,49),(42,58),(43,51),(44,60),(45,53),(46,62),(47,55),(48,64)], [(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 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E ··· 12J 12K 12L 16A 16B 16C 16D 16E 16F 16G 16H 24A ··· 24H 24I ··· 24T 24U 24V 24W 24X 48A ··· 48AF 48AG ··· 48AN order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 6 6 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 16 16 16 16 16 16 16 16 24 ··· 24 24 ··· 24 24 24 24 24 48 ··· 48 48 ··· 48 size 1 1 6 1 1 2 2 2 1 1 6 1 1 2 2 2 6 6 1 1 1 1 6 6 1 1 1 1 2 ··· 2 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 1 1 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 C8 C12 C12 C24 C24 S3 D6 C3×S3 C4×S3 M5(2) S3×C6 S3×C8 S3×C12 D6.C8 C3×M5(2) S3×C24 C3×D6.C8 kernel C3×D6.C8 C3×C3⋊C16 C3×C48 S3×C24 D6.C8 C3×C3⋊C8 S3×C12 C3⋊C16 C48 S3×C8 C3×Dic3 S3×C6 C3⋊C8 C4×S3 Dic3 D6 C48 C24 C16 C12 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 1 1 2 2 4 2 4 4 8 8 8 16

Matrix representation of C3×D6.C8 in GL2(𝔽97) generated by

 61 0 0 61
,
 62 0 0 36
,
 0 36 62 0
,
 12 0 0 85
G:=sub<GL(2,GF(97))| [61,0,0,61],[62,0,0,36],[0,62,36,0],[12,0,0,85] >;

C3×D6.C8 in GAP, Magma, Sage, TeX

C_3\times D_6.C_8
% in TeX

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

G:=SmallGroup(288,232);
// by ID

G=gap.SmallGroup(288,232);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,701,92,80,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^2=1,d^8=b^3,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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