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## G = C3×D6⋊Q8order 288 = 25·32

### Direct product of C3 and D6⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×D6⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×D6⋊Q8
 Lower central C3 — C2×C6 — C3×D6⋊Q8
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×D6⋊Q8
G = < a,b,c,d,e | a3=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b4c, ede-1=d-1 >

Subgroups: 370 in 161 conjugacy classes, 66 normal (58 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4 [×3], C2×C4 [×5], Q8 [×2], C23, C32, Dic3 [×2], Dic3 [×2], C12 [×14], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C3×S3 [×2], C3×C6 [×3], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×C12 [×6], C2×C12 [×8], C3×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×2], C3×Dic3 [×2], C3×C12 [×3], S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4, C22×C12, C6×Q8, C3×Dic6 [×2], S3×C12 [×2], C6×Dic3 [×3], C6×C12 [×3], S3×C2×C6, D6⋊Q8, C3×C22⋊Q8, C3×Dic3⋊C4 [×2], C3×D6⋊C4 [×2], C32×C4⋊C4, C6×Dic6, S3×C2×C12, C3×D6⋊Q8
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×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, S3×C6 [×3], C4○D12, S3×D4, S3×Q8, C6×D4, C6×Q8, C3×C4○D4, S3×C2×C6, D6⋊Q8, C3×C22⋊Q8, C3×C4○D12, C3×S3×D4, C3×S3×Q8, C3×D6⋊Q8

Smallest permutation representation of C3×D6⋊Q8
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 57)(2 56)(3 55)(4 60)(5 59)(6 58)(7 43)(8 48)(9 47)(10 46)(11 45)(12 44)(13 65)(14 64)(15 63)(16 62)(17 61)(18 66)(19 74)(20 73)(21 78)(22 77)(23 76)(24 75)(25 69)(26 68)(27 67)(28 72)(29 71)(30 70)(31 86)(32 85)(33 90)(34 89)(35 88)(36 87)(37 81)(38 80)(39 79)(40 84)(41 83)(42 82)(49 92)(50 91)(51 96)(52 95)(53 94)(54 93)
(1 74 14 67)(2 73 15 72)(3 78 16 71)(4 77 17 70)(5 76 18 69)(6 75 13 68)(7 40 91 32)(8 39 92 31)(9 38 93 36)(10 37 94 35)(11 42 95 34)(12 41 96 33)(19 63 27 56)(20 62 28 55)(21 61 29 60)(22 66 30 59)(23 65 25 58)(24 64 26 57)(43 83 50 90)(44 82 51 89)(45 81 52 88)(46 80 53 87)(47 79 54 86)(48 84 49 85)
(1 89 14 82)(2 88 15 81)(3 87 16 80)(4 86 17 79)(5 85 18 84)(6 90 13 83)(7 20 91 28)(8 19 92 27)(9 24 93 26)(10 23 94 25)(11 22 95 30)(12 21 96 29)(31 63 39 56)(32 62 40 55)(33 61 41 60)(34 66 42 59)(35 65 37 58)(36 64 38 57)(43 75 50 68)(44 74 51 67)(45 73 52 72)(46 78 53 71)(47 77 54 70)(48 76 49 69)

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

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

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

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×Q8 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 S3×D4 S3×Q8 C3×S3×D4 C3×S3×Q8 kernel C3×D6⋊Q8 C3×Dic3⋊C4 C3×D6⋊C4 C32×C4⋊C4 C6×Dic6 S3×C2×C12 D6⋊Q8 Dic3⋊C4 D6⋊C4 C3×C4⋊C4 C2×Dic6 S3×C2×C4 C3×C4⋊C4 C3×Dic3 S3×C6 C2×C12 C3×C6 C4⋊C4 Dic3 D6 C2×C4 C6 C6 C2 C6 C6 C2 C2 # reps 1 2 2 1 1 1 2 4 4 2 2 2 1 2 2 3 2 2 4 4 6 4 4 8 1 1 2 2

Matrix representation of C3×D6⋊Q8 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 4 3 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 4 3 0 0 8 9 0 0 0 0 1 0 0 0 0 1
,
 12 1 0 0 11 1 0 0 0 0 0 1 0 0 12 0
,
 8 0 0 0 3 5 0 0 0 0 10 9 0 0 9 3
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,3,10,0,0,0,0,1,0,0,0,0,1],[4,8,0,0,3,9,0,0,0,0,1,0,0,0,0,1],[12,11,0,0,1,1,0,0,0,0,0,12,0,0,1,0],[8,3,0,0,0,5,0,0,0,0,10,9,0,0,9,3] >;

C3×D6⋊Q8 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes Q_8
% in TeX

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

G:=SmallGroup(288,667);
// by ID

G=gap.SmallGroup(288,667);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,176,1598,555,268,9414]);
// 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=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^4*c,e*d*e^-1=d^-1>;
// generators/relations

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