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## G = C2×Q16⋊S3order 192 = 26·3

### Direct product of C2 and Q16⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Q16⋊S3
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×Q8 — C2×Q16⋊S3
 Lower central C3 — C6 — C12 — C2×Q16⋊S3
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C2×Q16⋊S3
G = < a,b,c,d,e | a2=b8=d3=e2=1, c2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 664 in 258 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×S3, C2×M4(2), C2×SD16, C2×Q16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C8⋊S3, C24⋊C2, C2×C3⋊C8, Q82S3, C3⋊Q16, C2×C24, C3×Q16, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, S3×Q8, S3×Q8, Q83S3, Q83S3, C6×Q8, C2×C8.C22, C2×C8⋊S3, C2×C24⋊C2, Q16⋊S3, C2×Q82S3, C2×C3⋊Q16, C6×Q16, C2×S3×Q8, C2×Q83S3, C2×Q16⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C8.C22, C22×D4, S3×D4, S3×C23, C2×C8.C22, Q16⋊S3, C2×S3×D4, C2×Q16⋊S3

Smallest permutation representation of C2×Q16⋊S3
On 96 points
Generators in S96
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 81)(40 82)(41 71)(42 72)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(57 77)(58 78)(59 79)(60 80)(61 73)(62 74)(63 75)(64 76)
(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 54 5 50)(2 53 6 49)(3 52 7 56)(4 51 8 55)(9 24 13 20)(10 23 14 19)(11 22 15 18)(12 21 16 17)(25 35 29 39)(26 34 30 38)(27 33 31 37)(28 40 32 36)(41 77 45 73)(42 76 46 80)(43 75 47 79)(44 74 48 78)(57 67 61 71)(58 66 62 70)(59 65 63 69)(60 72 64 68)(81 89 85 93)(82 96 86 92)(83 95 87 91)(84 94 88 90)
(1 84 79)(2 85 80)(3 86 73)(4 87 74)(5 88 75)(6 81 76)(7 82 77)(8 83 78)(9 38 63)(10 39 64)(11 40 57)(12 33 58)(13 34 59)(14 35 60)(15 36 61)(16 37 62)(17 27 70)(18 28 71)(19 29 72)(20 30 65)(21 31 66)(22 32 67)(23 25 68)(24 26 69)(41 52 92)(42 53 93)(43 54 94)(44 55 95)(45 56 96)(46 49 89)(47 50 90)(48 51 91)
(1 50)(2 55)(3 52)(4 49)(5 54)(6 51)(7 56)(8 53)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)(25 62)(26 59)(27 64)(28 61)(29 58)(30 63)(31 60)(32 57)(33 72)(34 69)(35 66)(36 71)(37 68)(38 65)(39 70)(40 67)(41 86)(42 83)(43 88)(44 85)(45 82)(46 87)(47 84)(48 81)(73 92)(74 89)(75 94)(76 91)(77 96)(78 93)(79 90)(80 95)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 6 6 12 12 2 2 2 4 4 4 4 6 6 12 12 2 2 2 4 4 12 12 4 4 8 8 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 C8.C22 S3×D4 S3×D4 Q16⋊S3 kernel C2×Q16⋊S3 C2×C8⋊S3 C2×C24⋊C2 Q16⋊S3 C2×Q8⋊2S3 C2×C3⋊Q16 C6×Q16 C2×S3×Q8 C2×Q8⋊3S3 C2×Q16 C4×S3 C2×Dic3 C22×S3 C2×C8 Q16 C2×Q8 C6 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 1 2 1 1 1 4 2 2 1 1 4

Matrix representation of C2×Q16⋊S3 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 53 45 34 34 0 0 0 0 67 59 0 34 0 0 0 0 0 45 14 28 0 0 0 0 67 0 6 20 0 0 0 0 0 0 0 0 31 62 31 62 0 0 0 0 11 42 11 42 0 0 0 0 42 11 31 62 0 0 0 0 62 31 11 42
,
 0 0 1 72 0 0 0 0 72 72 2 1 0 0 0 0 0 0 1 0 0 0 0 0 72 0 1 0 0 0 0 0 0 0 0 0 37 1 50 0 0 0 0 0 72 36 0 50 0 0 0 0 50 0 36 72 0 0 0 0 0 50 1 37
,
 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 1 0 0 0 0 0 1 72 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72
,
 72 72 2 1 0 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 0 37 1 50 0 0 0 0 0 37 36 23 23 0 0 0 0 23 0 37 1 0 0 0 0 50 50 37 36

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[53,67,0,67,0,0,0,0,45,59,45,0,0,0,0,0,34,0,14,6,0,0,0,0,34,34,28,20,0,0,0,0,0,0,0,0,31,11,42,62,0,0,0,0,62,42,11,31,0,0,0,0,31,11,31,11,0,0,0,0,62,42,62,42],[0,72,0,72,0,0,0,0,0,72,0,0,0,0,0,0,1,2,1,1,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,37,72,50,0,0,0,0,0,1,36,0,50,0,0,0,0,50,0,36,1,0,0,0,0,0,50,72,37],[72,72,72,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[72,0,0,0,0,0,0,0,72,0,0,72,0,0,0,0,2,1,1,1,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,37,37,23,50,0,0,0,0,1,36,0,50,0,0,0,0,50,23,37,37,0,0,0,0,0,23,1,36] >;

C2×Q16⋊S3 in GAP, Magma, Sage, TeX

C_2\times Q_{16}\rtimes S_3
% in TeX

G:=Group("C2xQ16:S3");
// GroupNames label

G:=SmallGroup(192,1323);
// by ID

G=gap.SmallGroup(192,1323);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,1123,185,136,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=d^3=e^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

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