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## G = C3×C6.D8order 288 = 25·32

### Direct product of C3 and C6.D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C6.D8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C6×D12 — C3×C6.D8
 Lower central C3 — C6 — C12 — C3×C6.D8
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×C6.D8
G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 338 in 113 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C8, C2×C4, C2×C4, D4 [×3], C23, C32, C12 [×4], C12 [×6], D6 [×4], C2×C6 [×2], C2×C6 [×5], C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3×C6 [×3], C3⋊C8, C24, D12 [×2], D12, C2×C12 [×2], C2×C12 [×5], C3×D4 [×3], C22×S3, C22×C6, D4⋊C4, C3×C12 [×2], C3×C12, S3×C6 [×4], C62, C2×C3⋊C8, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×C24, C2×D12, C6×D4, C3×C3⋊C8, C3×D12 [×2], C3×D12, C6×C12, C6×C12, S3×C2×C6, C6.D8, C3×D4⋊C4, C6×C3⋊C8, C32×C4⋊C4, C6×D12, C3×C6.D8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, D8, SD16, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], D4⋊C4, S3×C6, D6⋊C4, D4⋊S3, Q82S3, C3×C22⋊C4, C3×D8, C3×SD16, S3×C12, C3×D12, C3×C3⋊D4, C6.D8, C3×D4⋊C4, C3×D6⋊C4, C3×D4⋊S3, C3×Q82S3, C3×C6.D8

Smallest permutation representation of C3×C6.D8
On 96 points
Generators in S96
(1 30 94)(2 31 95)(3 32 96)(4 25 89)(5 26 90)(6 27 91)(7 28 92)(8 29 93)(9 71 38)(10 72 39)(11 65 40)(12 66 33)(13 67 34)(14 68 35)(15 69 36)(16 70 37)(17 78 63)(18 79 64)(19 80 57)(20 73 58)(21 74 59)(22 75 60)(23 76 61)(24 77 62)(41 84 52)(42 85 53)(43 86 54)(44 87 55)(45 88 56)(46 81 49)(47 82 50)(48 83 51)
(1 57 30 19 94 80)(2 73 95 20 31 58)(3 59 32 21 96 74)(4 75 89 22 25 60)(5 61 26 23 90 76)(6 77 91 24 27 62)(7 63 28 17 92 78)(8 79 93 18 29 64)(9 83 71 51 38 48)(10 41 39 52 72 84)(11 85 65 53 40 42)(12 43 33 54 66 86)(13 87 67 55 34 44)(14 45 35 56 68 88)(15 81 69 49 36 46)(16 47 37 50 70 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)
(1 54)(2 11)(3 52)(4 9)(5 50)(6 15)(7 56)(8 13)(10 21)(12 19)(14 17)(16 23)(18 55)(20 53)(22 51)(24 49)(25 71)(26 47)(27 69)(28 45)(29 67)(30 43)(31 65)(32 41)(33 57)(34 93)(35 63)(36 91)(37 61)(38 89)(39 59)(40 95)(42 73)(44 79)(46 77)(48 75)(58 85)(60 83)(62 81)(64 87)(66 80)(68 78)(70 76)(72 74)(82 90)(84 96)(86 94)(88 92)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12Z 24A ··· 24H order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 12 12 1 1 2 2 2 2 2 4 4 1 ··· 1 2 ··· 2 12 12 12 12 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 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 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D4 D6 D8 SD16 C3×S3 C4×S3 D12 C3×D4 C3⋊D4 C3×D4 S3×C6 C3×D8 C3×SD16 S3×C12 C3×D12 C3×C3⋊D4 D4⋊S3 Q8⋊2S3 C3×D4⋊S3 C3×Q8⋊2S3 kernel C3×C6.D8 C6×C3⋊C8 C32×C4⋊C4 C6×D12 C6.D8 C3×D12 C2×C3⋊C8 C3×C4⋊C4 C2×D12 D12 C3×C4⋊C4 C3×C12 C62 C2×C12 C3×C6 C3×C6 C4⋊C4 C12 C12 C12 C2×C6 C2×C6 C2×C4 C6 C6 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 1 1 2 2

Matrix representation of C3×C6.D8 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 65 0 0 0 0 1 0 0 0 0 1
,
 0 46 0 0 27 0 0 0 0 0 16 57 0 0 16 16
,
 0 1 0 0 1 0 0 0 0 0 17 2 0 0 2 56
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,65,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,46,0,0,0,0,0,16,16,0,0,57,16],[0,1,0,0,1,0,0,0,0,0,17,2,0,0,2,56] >;

C3×C6.D8 in GAP, Magma, Sage, TeX

C_3\times C_6.D_8
% in TeX

G:=Group("C3xC6.D8");
// GroupNames label

G:=SmallGroup(288,243);
// by ID

G=gap.SmallGroup(288,243);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,2524,1271,102,9414]);
// Polycyclic

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

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