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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

C1C16 — C3×SD64
C1C2C4C8C16C48C3×Q32 — C3×SD64
C1C2C4C8C16 — C3×SD64
C1C6C12C24C48 — C3×SD64

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

16C2
8C4
8C22
16C6
4D4
4Q8
8C12
8C2×C6
2D8
2Q16
4C3×Q8
4C3×D4
2C3×Q16
2C3×D8

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 2A2B3A3B4A4B6A6B6C6D8A8B12A12B12C12D16A16B16C16D24A24B24C24D32A···32H48A···48H96A···96P
order122334466668812121212161616162424242432···3248···4896···96
size11161121611161622221616222222222···22···22···2

57 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C3C6C6C6D4D8C3×D4D16C3×D8SD64C3×D16C3×SD64
kernelC3×SD64C96C3×D16C3×Q32SD64C32D16Q32C24C12C8C6C4C3C2C1
# reps11112222122448816

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

3500
010
001
,
9600
06236
06162
,
9600
010
0096
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

Export

Subgroup lattice of C3×SD64 in TeX

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