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## G = C6×Q8⋊3S3order 288 = 25·32

### Direct product of C6 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×Q8⋊3S3
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C2×C12 — C6×Q8⋊3S3
 Lower central C3 — C6 — C6×Q8⋊3S3
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C6×Q83S3
G = < a,b,c,d,e | a6=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 714 in 347 conjugacy classes, 178 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×6], C4 [×2], C22, C22 [×12], S3 [×6], C6 [×2], C6 [×4], C6 [×9], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×2], C12 [×12], C12 [×8], D6 [×6], D6 [×6], C2×C6 [×2], C2×C6 [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×6], C3×C6, C3×C6 [×2], C4×S3 [×12], D12 [×12], C2×Dic3, C2×C12 [×6], C2×C12 [×16], C3×D4 [×12], C3×Q8 [×8], C3×Q8 [×4], C22×S3 [×3], C22×C6 [×3], C2×C4○D4, C3×Dic3 [×2], C3×C12 [×6], S3×C6 [×6], S3×C6 [×6], C62, S3×C2×C4 [×3], C2×D12 [×3], Q83S3 [×8], C22×C12 [×3], C6×D4 [×3], C6×Q8 [×2], C6×Q8, C3×C4○D4 [×8], S3×C12 [×12], C3×D12 [×12], C6×Dic3, C6×C12 [×3], Q8×C32 [×4], S3×C2×C6 [×3], C2×Q83S3, C6×C4○D4, S3×C2×C12 [×3], C6×D12 [×3], C3×Q83S3 [×8], Q8×C3×C6, C6×Q83S3
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C4○D4 [×2], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], C2×C4○D4, S3×C6 [×7], Q83S3 [×2], C3×C4○D4 [×2], S3×C23, C23×C6, S3×C2×C6 [×7], C2×Q83S3, C6×C4○D4, C3×Q83S3 [×2], S3×C22×C6, C6×Q83S3

Smallest permutation representation of C6×Q83S3
On 96 points
Generators in S96
(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 22 41 35)(2 23 42 36)(3 24 37 31)(4 19 38 32)(5 20 39 33)(6 21 40 34)(7 48 53 17)(8 43 54 18)(9 44 49 13)(10 45 50 14)(11 46 51 15)(12 47 52 16)(25 94 73 63)(26 95 74 64)(27 96 75 65)(28 91 76 66)(29 92 77 61)(30 93 78 62)(55 80 86 67)(56 81 87 68)(57 82 88 69)(58 83 89 70)(59 84 90 71)(60 79 85 72)
(1 47 41 16)(2 48 42 17)(3 43 37 18)(4 44 38 13)(5 45 39 14)(6 46 40 15)(7 36 53 23)(8 31 54 24)(9 32 49 19)(10 33 50 20)(11 34 51 21)(12 35 52 22)(25 69 73 82)(26 70 74 83)(27 71 75 84)(28 72 76 79)(29 67 77 80)(30 68 78 81)(55 92 86 61)(56 93 87 62)(57 94 88 63)(58 95 89 64)(59 96 90 65)(60 91 85 66)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 78)(8 73)(9 74)(10 75)(11 76)(12 77)(13 64)(14 65)(15 66)(16 61)(17 62)(18 63)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 54)(26 49)(27 50)(28 51)(29 52)(30 53)(31 82)(32 83)(33 84)(34 79)(35 80)(36 81)(37 88)(38 89)(39 90)(40 85)(41 86)(42 87)(43 94)(44 95)(45 96)(46 91)(47 92)(48 93)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 3C 3D 3E 4A ··· 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6O 6P ··· 6AA 12A ··· 12L 12M ··· 12T 12U ··· 12AL order 1 2 2 2 2 ··· 2 3 3 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 6 ··· 6 1 1 2 2 2 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 3 ··· 3 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C3×C4○D4 Q8⋊3S3 C3×Q8⋊3S3 kernel C6×Q8⋊3S3 S3×C2×C12 C6×D12 C3×Q8⋊3S3 Q8×C3×C6 C2×Q8⋊3S3 S3×C2×C4 C2×D12 Q8⋊3S3 C6×Q8 C6×Q8 C2×C12 C3×Q8 C3×C6 C2×Q8 C2×C4 Q8 C6 C6 C2 # reps 1 3 3 8 1 2 6 6 16 2 1 3 4 4 2 6 8 8 2 4

Matrix representation of C6×Q83S3 in GL5(𝔽13)

 4 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 2 0 0 0 12 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 6 5 0 0 0 3 7
,
 1 0 0 0 0 0 9 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 4 12 0 0 0 2 9

G:=sub<GL(5,GF(13))| [4,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,2,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,6,3,0,0,0,5,7],[1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,12,9] >;

C6×Q83S3 in GAP, Magma, Sage, TeX

C_6\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C6xQ8:3S3");
// GroupNames label

G:=SmallGroup(288,996);
// by ID

G=gap.SmallGroup(288,996);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,409,192,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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