Copied to
clipboard

G = S3×Dic12order 288 = 25·32

Direct product of S3 and Dic12

direct product, metabelian, supersoluble, monomial

Aliases: S3×Dic12, C24.22D6, D6.12D12, Dic3.3D12, Dic6.14D6, C8.8S32, C3⋊C8.23D6, C31(S3×Q16), C6.5(S3×D4), (S3×C8).1S3, (C3×S3)⋊1Q16, C6.5(C2×D12), (S3×C24).1C2, C323(C2×Q16), (S3×C6).19D4, (C4×S3).35D6, C31(C2×Dic12), C2.10(S3×D12), C325Q166C2, (C3×Dic12)⋊8C2, (S3×Dic6).1C2, C323Q161C2, (C3×C12).47C23, (C3×C24).21C22, (C3×Dic3).22D4, (S3×C12).43C22, C12.121(C22×S3), (C3×Dic6).1C22, C324Q8.1C22, C4.44(C2×S32), (C3×C6).31(C2×D4), (C3×C3⋊C8).28C22, SmallGroup(288,447)

Series: Derived Chief Lower central Upper central

C1C3×C12 — S3×Dic12
C1C3C32C3×C6C3×C12S3×C12S3×Dic6 — S3×Dic12
C32C3×C6C3×C12 — S3×Dic12
C1C2C4C8

Generators and relations for S3×Dic12
 G = < a,b,c,d | a3=b2=c24=1, d2=c12, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 498 in 129 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], Q8 [×6], C32, Dic3, Dic3 [×8], C12 [×2], C12 [×4], D6, C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], Dic6 [×10], C4×S3, C4×S3 [×2], C2×Dic3 [×2], C2×C12, C3×Q8 [×2], C2×Q16, C3×Dic3, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12, S3×C6, S3×C8, Dic12, Dic12 [×5], C3⋊Q16 [×2], C2×C24, C3×Q16, C2×Dic6 [×2], S3×Q8 [×2], C3×C3⋊C8, C3×C24, S3×Dic3 [×2], C322Q8 [×2], C3×Dic6 [×2], S3×C12, C324Q8 [×2], C2×Dic12, S3×Q16, C323Q16 [×2], S3×C24, C3×Dic12, C325Q16, S3×Dic6 [×2], S3×Dic12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], Q16 [×2], C2×D4, D12 [×2], C22×S3 [×2], C2×Q16, S32, Dic12 [×2], C2×D12, S3×D4, C2×S32, C2×Dic12, S3×Q16, S3×D12, S3×Dic12

Smallest permutation representation of S3×Dic12
On 96 points
Generators in S96
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 92)(26 93)(27 94)(28 95)(29 96)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)
(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 45 13 33)(2 44 14 32)(3 43 15 31)(4 42 16 30)(5 41 17 29)(6 40 18 28)(7 39 19 27)(8 38 20 26)(9 37 21 25)(10 36 22 48)(11 35 23 47)(12 34 24 46)(49 76 61 88)(50 75 62 87)(51 74 63 86)(52 73 64 85)(53 96 65 84)(54 95 66 83)(55 94 67 82)(56 93 68 81)(57 92 69 80)(58 91 70 79)(59 90 71 78)(60 89 72 77)

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

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

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

45 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H12I24A24B24C24D24E···24J24K24L24M24N
order12223334444446666688881212121212121212122424242424···2424242424
size113322426121236362246622662244466242422224···46666

45 irreducible representations

dim111111222222222222444444
type++++++++++++++-++-+++-+-
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D6Q16D12D12Dic12S32S3×D4C2×S32S3×Q16S3×D12S3×Dic12
kernelS3×Dic12C323Q16S3×C24C3×Dic12C325Q16S3×Dic6S3×C8Dic12C3×Dic3S3×C6C3⋊C8C24Dic6C4×S3C3×S3Dic3D6S3C8C6C4C3C2C1
# reps121112111112214228111224

Matrix representation of S3×Dic12 in GL6(𝔽73)

100000
010000
001000
000100
000001
00007272
,
100000
010000
001000
000100
000010
00007272
,
0410000
16410000
007700
00661400
000010
000001
,
36330000
16370000
0075900
00666600
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,7,66,0,0,0,0,7,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,16,0,0,0,0,33,37,0,0,0,0,0,0,7,66,0,0,0,0,59,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×Dic12 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{12}
% in TeX

G:=Group("S3xDic12");
// GroupNames label

G:=SmallGroup(288,447);
// by ID

G=gap.SmallGroup(288,447);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,142,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^24=1,d^2=c^12,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽