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G = C6×D16order 192 = 26·3

Direct product of C6 and D16

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C6×D16, C12.44D8, C24.68D4, C4810C22, C24.63C23, (C2×C16)⋊5C6, C162(C2×C6), (C2×D8)⋊6C6, D81(C2×C6), C8.9(C3×D4), C4.6(C3×D8), C4.7(C6×D4), (C2×C48)⋊12C2, (C6×D8)⋊20C2, C6.84(C2×D8), C2.12(C6×D8), (C2×C6).55D8, C8.3(C22×C6), C12.314(C2×D4), (C2×C12).426D4, (C3×D8)⋊17C22, C22.14(C3×D8), (C2×C24).404C22, (C2×C8).84(C2×C6), (C2×C4).82(C3×D4), SmallGroup(192,938)

Series: Derived Chief Lower central Upper central

C1C8 — C6×D16
C1C2C4C8C24C3×D8C3×D16 — C6×D16
C1C2C4C8 — C6×D16
C1C2×C6C2×C12C2×C24 — C6×D16

Generators and relations for C6×D16
 G = < a,b,c | a6=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 274 in 98 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, D4, C23, C12, C2×C6, C2×C6, C16, C2×C8, D8, D8, C2×D4, C24, C2×C12, C3×D4, C22×C6, C2×C16, D16, C2×D8, C48, C2×C24, C3×D8, C3×D8, C6×D4, C2×D16, C2×C48, C3×D16, C6×D8, C6×D16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, D16, C2×D8, C3×D8, C6×D4, C2×D16, C3×D16, C6×D8, C6×D16

Smallest permutation representation of C6×D16
On 96 points
Generators in S96
(1 79 35 61 21 89)(2 80 36 62 22 90)(3 65 37 63 23 91)(4 66 38 64 24 92)(5 67 39 49 25 93)(6 68 40 50 26 94)(7 69 41 51 27 95)(8 70 42 52 28 96)(9 71 43 53 29 81)(10 72 44 54 30 82)(11 73 45 55 31 83)(12 74 46 56 32 84)(13 75 47 57 17 85)(14 76 48 58 18 86)(15 77 33 59 19 87)(16 78 34 60 20 88)
(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 52)(2 51)(3 50)(4 49)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 82)(34 81)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B6A···6F6G···6N8A8B8C8D12A12B12C12D16A···16H24A···24H48A···48P
order1222222233446···66···688881212121216···1624···2448···48
size1111888811221···18···8222222222···22···22···2

66 irreducible representations

dim111111112222222222
type+++++++++
imageC1C2C2C2C3C6C6C6D4D4D8D8C3×D4C3×D4D16C3×D8C3×D8C3×D16
kernelC6×D16C2×C48C3×D16C6×D8C2×D16C2×C16D16C2×D8C24C2×C12C12C2×C6C8C2×C4C6C4C22C2
# reps1142228411222284416

Matrix representation of C6×D16 in GL4(𝔽97) generated by

96000
03500
0010
0001
,
96000
09600
002695
00226
,
96000
0100
00712
00226
G:=sub<GL(4,GF(97))| [96,0,0,0,0,35,0,0,0,0,1,0,0,0,0,1],[96,0,0,0,0,96,0,0,0,0,26,2,0,0,95,26],[96,0,0,0,0,1,0,0,0,0,71,2,0,0,2,26] >;

C6×D16 in GAP, Magma, Sage, TeX

C_6\times D_{16}
% in TeX

G:=Group("C6xD16");
// GroupNames label

G:=SmallGroup(192,938);
// by ID

G=gap.SmallGroup(192,938);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c|a^6=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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