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## G = C3×D6⋊3Q8order 288 = 25·32

### Direct product of C3 and D6⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×D6⋊3Q8
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×D6⋊3Q8
 Lower central C3 — C2×C6 — C3×D6⋊3Q8
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C3×D63Q8
G = < a,b,c,d,e | a3=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 362 in 167 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C32, Dic3 [×3], C12 [×4], C12 [×13], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×S3 [×2], C3×C6 [×3], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×8], C3×Q8 [×8], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×3], C3×C12 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], S3×C2×C4, C22×C12, C6×Q8 [×2], C6×Q8, S3×C12 [×2], C6×Dic3, C6×Dic3 [×2], C6×C12, C6×C12 [×2], Q8×C32 [×2], S3×C2×C6, D63Q8, C3×C22⋊Q8, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×D6⋊C4 [×2], S3×C2×C12, Q8×C3×C6, C3×D63Q8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, S3×C6 [×3], S3×Q8, Q83S3, C2×C3⋊D4, C6×D4, C6×Q8, C3×C4○D4, C3×C3⋊D4 [×2], S3×C2×C6, D63Q8, C3×C22⋊Q8, C3×S3×Q8, C3×Q83S3, C6×C3⋊D4, C3×D63Q8

Smallest permutation representation of C3×D63Q8
On 96 points
Generators in S96
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(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 64)(2 63)(3 62)(4 61)(5 66)(6 65)(7 44)(8 43)(9 48)(10 47)(11 46)(12 45)(13 56)(14 55)(15 60)(16 59)(17 58)(18 57)(19 75)(20 74)(21 73)(22 78)(23 77)(24 76)(25 68)(26 67)(27 72)(28 71)(29 70)(30 69)(31 87)(32 86)(33 85)(34 90)(35 89)(36 88)(37 80)(38 79)(39 84)(40 83)(41 82)(42 81)(49 91)(50 96)(51 95)(52 94)(53 93)(54 92)
(1 28 16 24)(2 29 17 19)(3 30 18 20)(4 25 13 21)(5 26 14 22)(6 27 15 23)(7 84 93 89)(8 79 94 90)(9 80 95 85)(10 81 96 86)(11 82 91 87)(12 83 92 88)(31 43 41 52)(32 44 42 53)(33 45 37 54)(34 46 38 49)(35 47 39 50)(36 48 40 51)(55 75 66 70)(56 76 61 71)(57 77 62 72)(58 78 63 67)(59 73 64 68)(60 74 65 69)
(1 40 16 36)(2 41 17 31)(3 42 18 32)(4 37 13 33)(5 38 14 34)(6 39 15 35)(7 74 93 69)(8 75 94 70)(9 76 95 71)(10 77 96 72)(11 78 91 67)(12 73 92 68)(19 52 29 43)(20 53 30 44)(21 54 25 45)(22 49 26 46)(23 50 27 47)(24 51 28 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 12A 12B 12C 12D 12E ··· 12Z 12AA 12AB 12AC 12AD 12AE 12AF 12AG 12AH order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 6 6 1 1 2 2 2 2 2 4 4 6 6 12 12 1 ··· 1 2 ··· 2 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + - + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 Q8 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 C3×Q8 S3×C6 C3×C4○D4 C3×C3⋊D4 S3×Q8 Q8⋊3S3 C3×S3×Q8 C3×Q8⋊3S3 kernel C3×D6⋊3Q8 C3×Dic3⋊C4 C3×C4⋊Dic3 C3×D6⋊C4 S3×C2×C12 Q8×C3×C6 D6⋊3Q8 Dic3⋊C4 C4⋊Dic3 D6⋊C4 S3×C2×C4 C6×Q8 C6×Q8 C3×C12 S3×C6 C2×C12 C3×C6 C2×Q8 C12 C12 D6 C2×C4 C6 C4 C6 C6 C2 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 1 2 2 3 2 2 4 4 4 6 4 8 1 1 2 2

Matrix representation of C3×D63Q8 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 10 0 0 0 0 12 0 0 0 0 12
,
 0 10 0 0 4 0 0 0 0 0 12 0 0 0 8 1
,
 12 0 0 0 0 1 0 0 0 0 12 3 0 0 8 1
,
 12 0 0 0 0 12 0 0 0 0 5 0 0 0 12 8
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,12,0,0,0,0,12],[0,4,0,0,10,0,0,0,0,0,12,8,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,12,8,0,0,3,1],[12,0,0,0,0,12,0,0,0,0,5,12,0,0,0,8] >;

C3×D63Q8 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes_3Q_8
% in TeX

G:=Group("C3xD6:3Q8");
// GroupNames label

G:=SmallGroup(288,717);
// by ID

G=gap.SmallGroup(288,717);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,268,9414]);
// Polycyclic

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

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