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## G = C3×SD64order 192 = 26·3

### Direct product of C3 and SD64

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C3×SD64, C322C6, C964C2, D16.C6, Q321C6, C12.40D8, C6.16D16, C24.65D4, C48.20C22, C8.6(C3×D4), C4.2(C3×D8), C16.3(C2×C6), (C3×Q32)⋊5C2, C2.4(C3×D16), (C3×D16).2C2, SmallGroup(192,178)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C16 — C3×SD64
 Chief series C1 — C2 — C4 — C8 — C16 — C48 — C3×Q32 — C3×SD64
 Lower central C1 — C2 — C4 — C8 — C16 — C3×SD64
 Upper central C1 — C6 — C12 — C24 — C48 — C3×SD64

Generators and relations for C3×SD64
G = < a,b,c | a3=b32=c2=1, ab=ba, ac=ca, cbc=b15 >

Smallest permutation representation of C3×SD64
On 96 points
Generators in S96
(1 68 46)(2 69 47)(3 70 48)(4 71 49)(5 72 50)(6 73 51)(7 74 52)(8 75 53)(9 76 54)(10 77 55)(11 78 56)(12 79 57)(13 80 58)(14 81 59)(15 82 60)(16 83 61)(17 84 62)(18 85 63)(19 86 64)(20 87 33)(21 88 34)(22 89 35)(23 90 36)(24 91 37)(25 92 38)(26 93 39)(27 94 40)(28 95 41)(29 96 42)(30 65 43)(31 66 44)(32 67 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)
(2 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 43)(34 58)(35 41)(36 56)(37 39)(38 54)(40 52)(42 50)(44 48)(45 63)(47 61)(49 59)(51 57)(53 55)(60 64)(65 87)(66 70)(67 85)(69 83)(71 81)(72 96)(73 79)(74 94)(75 77)(76 92)(78 90)(80 88)(82 86)(89 95)(91 93)

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

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

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

57 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 16A 16B 16C 16D 24A 24B 24C 24D 32A ··· 32H 48A ··· 48H 96A ··· 96P order 1 2 2 3 3 4 4 6 6 6 6 8 8 12 12 12 12 16 16 16 16 24 24 24 24 32 ··· 32 48 ··· 48 96 ··· 96 size 1 1 16 1 1 2 16 1 1 16 16 2 2 2 2 16 16 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D8 C3×D4 D16 C3×D8 SD64 C3×D16 C3×SD64 kernel C3×SD64 C96 C3×D16 C3×Q32 SD64 C32 D16 Q32 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 4 4 8 8 16

Matrix representation of C3×SD64 in GL3(𝔽97) generated by

 35 0 0 0 1 0 0 0 1
,
 96 0 0 0 62 36 0 61 62
,
 96 0 0 0 1 0 0 0 96
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[96,0,0,0,62,61,0,36,62],[96,0,0,0,1,0,0,0,96] >;

C3×SD64 in GAP, Magma, Sage, TeX

C_3\times {\rm SD}_{64}
% in TeX

G:=Group("C3xSD64");
// GroupNames label

G:=SmallGroup(192,178);
// by ID

G=gap.SmallGroup(192,178);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,672,197,1011,514,192,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c|a^3=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^15>;
// generators/relations

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