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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

Aliases: C6×Q16, C12.43D4, C24.22C22, C12.46C23, (C2×C8).4C6, C8.5(C2×C6), C4.8(C3×D4), C2.13(C6×D4), C6.76(C2×D4), (C2×C6).54D4, Q8.4(C2×C6), (C2×Q8).6C6, (C6×Q8).9C2, (C2×C24).10C2, C4.3(C22×C6), C22.16(C3×D4), (C3×Q8).12C22, (C2×C12).131C22, (C2×C4).27(C2×C6), SmallGroup(96,181)

Series: Derived Chief Lower central Upper central

C1C4 — C6×Q16
C1C2C4C12C3×Q8C3×Q16 — C6×Q16
C1C2C4 — C6×Q16
C1C2×C6C2×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)])

C6×Q16 is a maximal subgroup of
C6.5Q32  Q16.Dic3  C24.27C23  Dic33Q16  C24.26D4  Q16⋊Dic3  (C2×Q16)⋊S3  D65Q16  D12.17D4  D63Q16  C24.36D4  C24.37D4  C24.28D4  C24.29D4  D12.30D4

42 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F6A···6F8A8B8C8D12A12B12C12D12E···12L24A···24H
order1222334444446···688881212121212···1224···24
size1111112244441···1222222224···42···2

42 irreducible representations

dim11111111222222
type++++++-
imageC1C2C2C2C3C6C6C6D4D4Q16C3×D4C3×D4C3×Q16
kernelC6×Q16C2×C24C3×Q16C6×Q8C2×Q16C2×C8Q16C2×Q8C12C2×C6C6C4C22C2
# reps11422284114228

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

900
0720
0072
,
100
05716
05757
,
7200
02154
05452
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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