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## G = C3×C8⋊Dic3order 288 = 25·32

### Direct product of C3 and C8⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C8⋊Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4⋊Dic3 — C3×C8⋊Dic3
 Lower central C3 — C6 — C12 — C3×C8⋊Dic3
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C3×C8⋊Dic3
G = < a,b,c,d | a3=b8=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >

Subgroups: 186 in 83 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C3×C6, C24, C24, C2×Dic3, C2×C12, C2×C12, C4.Q8, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×C24, C3×C24, C6×Dic3, C6×C12, C8⋊Dic3, C3×C4.Q8, C3×C4⋊Dic3, C6×C24, C3×C8⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, SD16, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C4.Q8, C3×Dic3, S3×C6, C24⋊C2, C4⋊Dic3, C3×C4⋊C4, C3×SD16, C3×Dic6, C3×D12, C6×Dic3, C8⋊Dic3, C3×C4.Q8, C3×C24⋊C2, C3×C4⋊Dic3, C3×C8⋊Dic3

Smallest permutation representation of C3×C8⋊Dic3
On 96 points
Generators in S96
(1 77 64)(2 78 57)(3 79 58)(4 80 59)(5 73 60)(6 74 61)(7 75 62)(8 76 63)(9 85 43)(10 86 44)(11 87 45)(12 88 46)(13 81 47)(14 82 48)(15 83 41)(16 84 42)(17 67 31)(18 68 32)(19 69 25)(20 70 26)(21 71 27)(22 72 28)(23 65 29)(24 66 30)(33 93 53)(34 94 54)(35 95 55)(36 96 56)(37 89 49)(38 90 50)(39 91 51)(40 92 52)
(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 66 64 24 77 30)(2 67 57 17 78 31)(3 68 58 18 79 32)(4 69 59 19 80 25)(5 70 60 20 73 26)(6 71 61 21 74 27)(7 72 62 22 75 28)(8 65 63 23 76 29)(9 90 85 50 43 38)(10 91 86 51 44 39)(11 92 87 52 45 40)(12 93 88 53 46 33)(13 94 81 54 47 34)(14 95 82 55 48 35)(15 96 83 56 41 36)(16 89 84 49 42 37)
(1 50 24 9)(2 53 17 12)(3 56 18 15)(4 51 19 10)(5 54 20 13)(6 49 21 16)(7 52 22 11)(8 55 23 14)(25 44 59 91)(26 47 60 94)(27 42 61 89)(28 45 62 92)(29 48 63 95)(30 43 64 90)(31 46 57 93)(32 41 58 96)(33 67 88 78)(34 70 81 73)(35 65 82 76)(36 68 83 79)(37 71 84 74)(38 66 85 77)(39 69 86 80)(40 72 87 75)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12X 24A ··· 24AF order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 2 2 2 2 2 12 12 12 12 1 ··· 1 2 ··· 2 2 2 2 2 2 ··· 2 12 ··· 12 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 C3 C4 C6 C6 C12 S3 Q8 D4 Dic3 D6 SD16 C3×S3 Dic6 C3×Q8 D12 C3×D4 C3×Dic3 S3×C6 C24⋊C2 C3×SD16 C3×Dic6 C3×D12 C3×C24⋊C2 kernel C3×C8⋊Dic3 C3×C4⋊Dic3 C6×C24 C8⋊Dic3 C3×C24 C4⋊Dic3 C2×C24 C24 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 2 1 2 4 4 2 8 1 1 1 2 1 4 2 2 2 2 2 4 2 8 8 4 4 16

Matrix representation of C3×C8⋊Dic3 in GL3(𝔽73) generated by

 8 0 0 0 8 0 0 0 8
,
 72 0 0 0 10 0 0 0 51
,
 72 0 0 0 8 0 0 0 64
,
 46 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[72,0,0,0,10,0,0,0,51],[72,0,0,0,8,0,0,0,64],[46,0,0,0,0,1,0,1,0] >;

C3×C8⋊Dic3 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes {\rm Dic}_3
% in TeX

G:=Group("C3xC8:Dic3");
// GroupNames label

G:=SmallGroup(288,251);
// by ID

G=gap.SmallGroup(288,251);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,176,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

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