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## G = C6×Q16order 96 = 25·3

### Direct product of C6 and Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×Q16
 Chief series C1 — C2 — C4 — C12 — C3×Q8 — C3×Q16 — C6×Q16
 Lower central C1 — C2 — C4 — C6×Q16
 Upper central C1 — C2×C6 — C2×C12 — C6×Q16

Generators and relations for C6×Q16
G = < a,b,c | a6=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 60 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×4], Q8 [×2], C12 [×2], C12 [×4], C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C3×Q8 [×2], C2×Q16, C2×C24, C3×Q16 [×4], C6×Q8 [×2], C6×Q16
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], Q16 [×2], C2×D4, C3×D4 [×2], C22×C6, C2×Q16, C3×Q16 [×2], C6×D4, C6×Q16

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L 24A ··· 24H order 1 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 2 2 2 2 4 ··· 4 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 Q16 C3×D4 C3×D4 C3×Q16 kernel C6×Q16 C2×C24 C3×Q16 C6×Q8 C2×Q16 C2×C8 Q16 C2×Q8 C12 C2×C6 C6 C4 C22 C2 # reps 1 1 4 2 2 2 8 4 1 1 4 2 2 8

Matrix representation of C6×Q16 in GL3(𝔽73) generated by

 9 0 0 0 72 0 0 0 72
,
 1 0 0 0 57 16 0 57 57
,
 72 0 0 0 21 54 0 54 52
G:=sub<GL(3,GF(73))| [9,0,0,0,72,0,0,0,72],[1,0,0,0,57,57,0,16,57],[72,0,0,0,21,54,0,54,52] >;

C6×Q16 in GAP, Magma, Sage, TeX

C_6\times Q_{16}
% in TeX

G:=Group("C6xQ16");
// GroupNames label

G:=SmallGroup(96,181);
// by ID

G=gap.SmallGroup(96,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,295,2164,1090,88]);
// Polycyclic

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

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