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## G = C3×C6.Q16order 288 = 25·32

### Direct product of C3 and C6.Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C6.Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C6×C3⋊C8 — C3×C6.Q16
 Lower central C3 — C6 — C12 — C3×C6.Q16
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×C6.Q16
G = < a,b,c,d | a3=b12=c4=1, d2=b9c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b9c-1 >

Subgroups: 178 in 85 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3, C12 [×4], C12 [×7], C2×C6 [×2], C2×C6, C4⋊C4, C4⋊C4, C2×C8, C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C2.D8, C3×Dic3, C3×C12 [×2], C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C6×C12, C6×C12, C6.Q16, C3×C2.D8, C6×C3⋊C8, C3×C4⋊Dic3, C32×C4⋊C4, C3×C6.Q16
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, C12 [×2], D6, C2×C6, C4⋊C4, D8, Q16, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C2.D8, S3×C6, Dic3⋊C4, D4⋊S3, C3⋊Q16, C3×C4⋊C4, C3×D8, C3×Q16, C3×Dic6, S3×C12, C3×C3⋊D4, C6.Q16, C3×C2.D8, C3×Dic3⋊C4, C3×D4⋊S3, C3×C3⋊Q16, C3×C6.Q16

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

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

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

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

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 2 2 4 4 4 4 type + + + + + - + + + - - + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 Q8 D4 D6 D8 Q16 C3×S3 Dic6 C4×S3 C3×Q8 C3⋊D4 C3×D4 S3×C6 C3×D8 C3×Q16 C3×Dic6 S3×C12 C3×C3⋊D4 D4⋊S3 C3⋊Q16 C3×D4⋊S3 C3×C3⋊Q16 kernel C3×C6.Q16 C6×C3⋊C8 C3×C4⋊Dic3 C32×C4⋊C4 C6.Q16 C3×C3⋊C8 C2×C3⋊C8 C4⋊Dic3 C3×C4⋊C4 C3⋊C8 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.Q16 in GL5(𝔽73)

 64 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 9 0 0 0 0 41 65 0 0 0 0 0 0 1 0 0 0 72 0
,
 27 0 0 0 0 0 72 0 0 0 0 51 1 0 0 0 0 0 47 36 0 0 0 36 26
,
 72 0 0 0 0 0 69 7 0 0 0 8 4 0 0 0 0 0 57 57 0 0 0 16 57

G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,9,41,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,1,0],[27,0,0,0,0,0,72,51,0,0,0,0,1,0,0,0,0,0,47,36,0,0,0,36,26],[72,0,0,0,0,0,69,8,0,0,0,7,4,0,0,0,0,0,57,16,0,0,0,57,57] >;

C3×C6.Q16 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC6.Q16");
// GroupNames label

G:=SmallGroup(288,241);
// by ID

G=gap.SmallGroup(288,241);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,365,92,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=b^9*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^9*c^-1>;
// generators/relations

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