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G = C3×C8⋊Dic3order 288 = 25·32

Direct product of C3 and C8⋊Dic3

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C8⋊Dic3, C242C12, C246Dic3, C62.78D4, C12.31Dic6, (C3×C24)⋊8C4, C82(C3×Dic3), C12.4(C3×Q8), (C2×C24).12C6, (C6×C24).14C2, (C2×C24).34S3, (C2×C6).69D12, C4⋊Dic3.2C6, (C3×C12).22Q8, C4.4(C3×Dic6), C6.2(C3×SD16), C4.6(C6×Dic3), C12.41(C2×C12), C329(C4.Q8), (C2×C12).432D6, (C3×C6).16SD16, C22.8(C3×D12), C6.14(C24⋊C2), C6.18(C4⋊Dic3), C12.63(C2×Dic3), (C6×C12).316C22, C6.5(C3×C4⋊C4), C32(C3×C4.Q8), (C2×C8).6(C3×S3), C2.2(C3×C24⋊C2), (C2×C4).71(S3×C6), (C2×C6).17(C3×D4), C2.3(C3×C4⋊Dic3), (C3×C6).31(C4⋊C4), (C3×C12).127(C2×C4), (C2×C12).101(C2×C6), (C3×C4⋊Dic3).21C2, SmallGroup(288,251)

Series: Derived Chief Lower central Upper central

C1C12 — C3×C8⋊Dic3
C1C3C6C12C2×C12C6×C12C3×C4⋊Dic3 — C3×C8⋊Dic3
C3C6C12 — C3×C8⋊Dic3
C1C2×C6C2×C12C2×C24

Generators and relations for C3×C8⋊Dic3
 G = < a,b,c,d | a3=b8=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >

Subgroups: 186 in 83 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C4⋊C4 [×2], C2×C8, C3×C6, C3×C6 [×2], C24 [×4], C24 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C4.Q8, C3×Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C2×C24 [×2], C2×C24, C3×C24 [×2], C6×Dic3 [×2], C6×C12, C8⋊Dic3, C3×C4.Q8, C3×C4⋊Dic3 [×2], C6×C24, C3×C8⋊Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, SD16 [×2], C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C4.Q8, C3×Dic3 [×2], S3×C6, C24⋊C2 [×2], C4⋊Dic3, C3×C4⋊C4, C3×SD16 [×2], C3×Dic6, C3×D12, C6×Dic3, C8⋊Dic3, C3×C4.Q8, C3×C24⋊C2 [×2], C3×C4⋊Dic3, C3×C8⋊Dic3

Smallest permutation representation of C3×C8⋊Dic3
On 96 points
Generators in S96
(1 75 59)(2 76 60)(3 77 61)(4 78 62)(5 79 63)(6 80 64)(7 73 57)(8 74 58)(9 85 43)(10 86 44)(11 87 45)(12 88 46)(13 81 47)(14 82 48)(15 83 41)(16 84 42)(17 69 31)(18 70 32)(19 71 25)(20 72 26)(21 65 27)(22 66 28)(23 67 29)(24 68 30)(33 90 50)(34 91 51)(35 92 52)(36 93 53)(37 94 54)(38 95 55)(39 96 56)(40 89 49)
(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 68 59 24 75 30)(2 69 60 17 76 31)(3 70 61 18 77 32)(4 71 62 19 78 25)(5 72 63 20 79 26)(6 65 64 21 80 27)(7 66 57 22 73 28)(8 67 58 23 74 29)(9 90 85 50 43 33)(10 91 86 51 44 34)(11 92 87 52 45 35)(12 93 88 53 46 36)(13 94 81 54 47 37)(14 95 82 55 48 38)(15 96 83 56 41 39)(16 89 84 49 42 40)
(1 50 24 9)(2 53 17 12)(3 56 18 15)(4 51 19 10)(5 54 20 13)(6 49 21 16)(7 52 22 11)(8 55 23 14)(25 44 62 91)(26 47 63 94)(27 42 64 89)(28 45 57 92)(29 48 58 95)(30 43 59 90)(31 46 60 93)(32 41 61 96)(33 68 85 75)(34 71 86 78)(35 66 87 73)(36 69 88 76)(37 72 81 79)(38 67 82 74)(39 70 83 77)(40 65 84 80)

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

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

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

90 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6O8A8B8C8D12A···12P12Q···12X24A···24AF
order1222333334444446···66···6888812···1212···1224···24
size11111122222121212121···12···222222···212···122···2

90 irreducible representations

dim11111111222222222222222222
type++++-+-+-+
imageC1C2C2C3C4C6C6C12S3Q8D4Dic3D6SD16C3×S3Dic6C3×Q8D12C3×D4C3×Dic3S3×C6C24⋊C2C3×SD16C3×Dic6C3×D12C3×C24⋊C2
kernelC3×C8⋊Dic3C3×C4⋊Dic3C6×C24C8⋊Dic3C3×C24C4⋊Dic3C2×C24C24C2×C24C3×C12C62C24C2×C12C3×C6C2×C8C12C12C2×C6C2×C6C8C2×C4C6C6C4C22C2
# reps121244281112142222242884416

Matrix representation of C3×C8⋊Dic3 in GL3(𝔽73) generated by

800
080
008
,
7200
0100
0051
,
7200
080
0064
,
4600
001
010
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[72,0,0,0,10,0,0,0,51],[72,0,0,0,8,0,0,0,64],[46,0,0,0,0,1,0,1,0] >;

C3×C8⋊Dic3 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes {\rm Dic}_3
% in TeX

G:=Group("C3xC8:Dic3");
// GroupNames label

G:=SmallGroup(288,251);
// by ID

G=gap.SmallGroup(288,251);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,176,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽