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## G = C6×C3⋊Q16order 288 = 25·32

### Direct product of C6 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C6×C3⋊Q16
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×Dic6 — C6×Dic6 — C6×C3⋊Q16
 Lower central C3 — C6 — C12 — C6×C3⋊Q16
 Upper central C1 — C2×C6 — C2×C12 — C6×Q8

Generators and relations for C6×C3⋊Q16
G = < a,b,c,d | a6=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 266 in 139 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], Q8 [×4], C32, Dic3 [×2], C12 [×4], C12 [×12], C2×C6 [×2], C2×C6, C2×C8, Q16 [×4], C2×Q8, C2×Q8, C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12 [×2], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×9], C2×Q16, C3×Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C62, C2×C3⋊C8, C3⋊Q16 [×4], C2×C24, C3×Q16 [×4], C2×Dic6, C6×Q8 [×2], C6×Q8 [×2], C3×C3⋊C8 [×2], C3×Dic6 [×2], C3×Dic6, C6×Dic3, C6×C12, C6×C12, Q8×C32 [×2], Q8×C32, C2×C3⋊Q16, C6×Q16, C6×C3⋊C8, C3×C3⋊Q16 [×4], C6×Dic6, Q8×C3×C6, C6×C3⋊Q16
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], Q16 [×2], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×Q16, S3×C6 [×3], C3⋊Q16 [×2], C3×Q16 [×2], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, C2×C3⋊Q16, C6×Q16, C3×C3⋊Q16 [×2], C6×C3⋊D4, C6×C3⋊Q16

Smallest permutation representation of C6×C3⋊Q16
On 96 points
Generators in S96
(1 70 17 61 91 39)(2 71 18 62 92 40)(3 72 19 63 93 33)(4 65 20 64 94 34)(5 66 21 57 95 35)(6 67 22 58 96 36)(7 68 23 59 89 37)(8 69 24 60 90 38)(9 53 85 26 76 42)(10 54 86 27 77 43)(11 55 87 28 78 44)(12 56 88 29 79 45)(13 49 81 30 80 46)(14 50 82 31 73 47)(15 51 83 32 74 48)(16 52 84 25 75 41)
(1 17 91)(2 92 18)(3 19 93)(4 94 20)(5 21 95)(6 96 22)(7 23 89)(8 90 24)(9 85 76)(10 77 86)(11 87 78)(12 79 88)(13 81 80)(14 73 82)(15 83 74)(16 75 84)(25 52 41)(26 42 53)(27 54 43)(28 44 55)(29 56 45)(30 46 49)(31 50 47)(32 48 51)(33 72 63)(34 64 65)(35 66 57)(36 58 67)(37 68 59)(38 60 69)(39 70 61)(40 62 71)
(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 80 5 76)(2 79 6 75)(3 78 7 74)(4 77 8 73)(9 17 13 21)(10 24 14 20)(11 23 15 19)(12 22 16 18)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 71 45 67)(42 70 46 66)(43 69 47 65)(44 68 48 72)(49 57 53 61)(50 64 54 60)(51 63 55 59)(52 62 56 58)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)

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

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

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

72 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 12B 12C 12D 12E ··· 12Z 12AA 12AB 12AC 12AD 24A ··· 24H 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 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 1 1 2 2 2 2 2 4 4 12 12 1 ··· 1 2 ··· 2 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 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 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D4 D6 D6 Q16 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×Q16 C3×C3⋊D4 C3×C3⋊D4 C3⋊Q16 C3×C3⋊Q16 kernel C6×C3⋊Q16 C6×C3⋊C8 C3×C3⋊Q16 C6×Dic6 Q8×C3×C6 C2×C3⋊Q16 C2×C3⋊C8 C3⋊Q16 C2×Dic6 C6×Q8 C6×Q8 C3×C12 C62 C2×C12 C3×Q8 C3×C6 C2×Q8 C12 C12 C2×C6 C2×C6 C2×C4 Q8 C6 C4 C22 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 1 1 1 2 4 2 2 2 2 2 2 4 8 4 4 2 4

Matrix representation of C6×C3⋊Q16 in GL5(𝔽73)

 65 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 64 9 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 72 44 0 0 0 63 1 0 0 0 0 0 0 58 0 0 0 39 32
,
 1 0 0 0 0 0 72 44 0 0 0 0 1 0 0 0 0 0 32 49 0 0 0 64 41

G:=sub<GL(5,GF(73))| [65,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,64,0,0,0,0,9,8,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,72,63,0,0,0,44,1,0,0,0,0,0,0,39,0,0,0,58,32],[1,0,0,0,0,0,72,0,0,0,0,44,1,0,0,0,0,0,32,64,0,0,0,49,41] >;

C6×C3⋊Q16 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes Q_{16}
% in TeX

G:=Group("C6xC3:Q16");
// GroupNames label

G:=SmallGroup(288,714);
// by ID

G=gap.SmallGroup(288,714);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,268,2524,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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