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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 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, C8, C2×C4, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C16, C16, C2×C8, C3×S3, C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, M5(2), C3×Dic3, C3×C12, S3×C6, C3⋊C16, C48 [×2], C48 [×2], 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 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, C12 [×2], D6, C2×C6, C2×C8, C3×S3, C24 [×2], 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 61 39)(2 62 40)(3 63 41)(4 64 42)(5 49 43)(6 50 44)(7 51 45)(8 52 46)(9 53 47)(10 54 48)(11 55 33)(12 56 34)(13 57 35)(14 58 36)(15 59 37)(16 60 38)(17 83 78)(18 84 79)(19 85 80)(20 86 65)(21 87 66)(22 88 67)(23 89 68)(24 90 69)(25 91 70)(26 92 71)(27 93 72)(28 94 73)(29 95 74)(30 96 75)(31 81 76)(32 82 77)
(1 47 61 9 39 53)(2 48 62 10 40 54)(3 33 63 11 41 55)(4 34 64 12 42 56)(5 35 49 13 43 57)(6 36 50 14 44 58)(7 37 51 15 45 59)(8 38 52 16 46 60)(17 91 78 25 83 70)(18 92 79 26 84 71)(19 93 80 27 85 72)(20 94 65 28 86 73)(21 95 66 29 87 74)(22 96 67 30 88 75)(23 81 68 31 89 76)(24 82 69 32 90 77)
(1 76)(2 69)(3 78)(4 71)(5 80)(6 73)(7 66)(8 75)(9 68)(10 77)(11 70)(12 79)(13 72)(14 65)(15 74)(16 67)(17 63)(18 56)(19 49)(20 58)(21 51)(22 60)(23 53)(24 62)(25 55)(26 64)(27 57)(28 50)(29 59)(30 52)(31 61)(32 54)(33 91)(34 84)(35 93)(36 86)(37 95)(38 88)(39 81)(40 90)(41 83)(42 92)(43 85)(44 94)(45 87)(46 96)(47 89)(48 82)
(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)| (1,61,39)(2,62,40)(3,63,41)(4,64,42)(5,49,43)(6,50,44)(7,51,45)(8,52,46)(9,53,47)(10,54,48)(11,55,33)(12,56,34)(13,57,35)(14,58,36)(15,59,37)(16,60,38)(17,83,78)(18,84,79)(19,85,80)(20,86,65)(21,87,66)(22,88,67)(23,89,68)(24,90,69)(25,91,70)(26,92,71)(27,93,72)(28,94,73)(29,95,74)(30,96,75)(31,81,76)(32,82,77), (1,47,61,9,39,53)(2,48,62,10,40,54)(3,33,63,11,41,55)(4,34,64,12,42,56)(5,35,49,13,43,57)(6,36,50,14,44,58)(7,37,51,15,45,59)(8,38,52,16,46,60)(17,91,78,25,83,70)(18,92,79,26,84,71)(19,93,80,27,85,72)(20,94,65,28,86,73)(21,95,66,29,87,74)(22,96,67,30,88,75)(23,81,68,31,89,76)(24,82,69,32,90,77), (1,76)(2,69)(3,78)(4,71)(5,80)(6,73)(7,66)(8,75)(9,68)(10,77)(11,70)(12,79)(13,72)(14,65)(15,74)(16,67)(17,63)(18,56)(19,49)(20,58)(21,51)(22,60)(23,53)(24,62)(25,55)(26,64)(27,57)(28,50)(29,59)(30,52)(31,61)(32,54)(33,91)(34,84)(35,93)(36,86)(37,95)(38,88)(39,81)(40,90)(41,83)(42,92)(43,85)(44,94)(45,87)(46,96)(47,89)(48,82), (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( (1,61,39)(2,62,40)(3,63,41)(4,64,42)(5,49,43)(6,50,44)(7,51,45)(8,52,46)(9,53,47)(10,54,48)(11,55,33)(12,56,34)(13,57,35)(14,58,36)(15,59,37)(16,60,38)(17,83,78)(18,84,79)(19,85,80)(20,86,65)(21,87,66)(22,88,67)(23,89,68)(24,90,69)(25,91,70)(26,92,71)(27,93,72)(28,94,73)(29,95,74)(30,96,75)(31,81,76)(32,82,77), (1,47,61,9,39,53)(2,48,62,10,40,54)(3,33,63,11,41,55)(4,34,64,12,42,56)(5,35,49,13,43,57)(6,36,50,14,44,58)(7,37,51,15,45,59)(8,38,52,16,46,60)(17,91,78,25,83,70)(18,92,79,26,84,71)(19,93,80,27,85,72)(20,94,65,28,86,73)(21,95,66,29,87,74)(22,96,67,30,88,75)(23,81,68,31,89,76)(24,82,69,32,90,77), (1,76)(2,69)(3,78)(4,71)(5,80)(6,73)(7,66)(8,75)(9,68)(10,77)(11,70)(12,79)(13,72)(14,65)(15,74)(16,67)(17,63)(18,56)(19,49)(20,58)(21,51)(22,60)(23,53)(24,62)(25,55)(26,64)(27,57)(28,50)(29,59)(30,52)(31,61)(32,54)(33,91)(34,84)(35,93)(36,86)(37,95)(38,88)(39,81)(40,90)(41,83)(42,92)(43,85)(44,94)(45,87)(46,96)(47,89)(48,82), (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([(1,61,39),(2,62,40),(3,63,41),(4,64,42),(5,49,43),(6,50,44),(7,51,45),(8,52,46),(9,53,47),(10,54,48),(11,55,33),(12,56,34),(13,57,35),(14,58,36),(15,59,37),(16,60,38),(17,83,78),(18,84,79),(19,85,80),(20,86,65),(21,87,66),(22,88,67),(23,89,68),(24,90,69),(25,91,70),(26,92,71),(27,93,72),(28,94,73),(29,95,74),(30,96,75),(31,81,76),(32,82,77)], [(1,47,61,9,39,53),(2,48,62,10,40,54),(3,33,63,11,41,55),(4,34,64,12,42,56),(5,35,49,13,43,57),(6,36,50,14,44,58),(7,37,51,15,45,59),(8,38,52,16,46,60),(17,91,78,25,83,70),(18,92,79,26,84,71),(19,93,80,27,85,72),(20,94,65,28,86,73),(21,95,66,29,87,74),(22,96,67,30,88,75),(23,81,68,31,89,76),(24,82,69,32,90,77)], [(1,76),(2,69),(3,78),(4,71),(5,80),(6,73),(7,66),(8,75),(9,68),(10,77),(11,70),(12,79),(13,72),(14,65),(15,74),(16,67),(17,63),(18,56),(19,49),(20,58),(21,51),(22,60),(23,53),(24,62),(25,55),(26,64),(27,57),(28,50),(29,59),(30,52),(31,61),(32,54),(33,91),(34,84),(35,93),(36,86),(37,95),(38,88),(39,81),(40,90),(41,83),(42,92),(43,85),(44,94),(45,87),(46,96),(47,89),(48,82)], [(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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