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G = C3×C122Q8order 288 = 25·32

Direct product of C3 and C122Q8

direct product, metabelian, supersoluble, monomial

Aliases: C3×C122Q8, C128Dic6, C12.65D12, C122.8C2, C62.160C23, (C3×C12)⋊9Q8, C122(C3×Q8), C6.1(C6×D4), C6.2(C6×Q8), (C4×C12).8C6, C42(C3×Dic6), C4.4(C3×D12), C2.4(C6×D12), C329(C4⋊Q8), (C4×C12).24S3, C12.27(C3×D4), C6.89(C2×D12), C4⋊Dic3.4C6, C42.4(C3×S3), C2.4(C6×Dic6), (C2×C12).437D6, (C3×C12).129D4, (C2×Dic6).2C6, C6.48(C2×Dic6), (C6×Dic6).17C2, (C6×C12).319C22, (C6×Dic3).87C22, C31(C3×C4⋊Q8), (C2×C4).75(S3×C6), C22.34(S3×C2×C6), (C3×C6).45(C2×Q8), (C3×C6).172(C2×D4), (C2×C12).104(C2×C6), (C3×C4⋊Dic3).23C2, (C2×C6).15(C22×C6), (C2×Dic3).1(C2×C6), (C2×C6).293(C22×S3), SmallGroup(288,640)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C122Q8
C1C3C6C2×C6C62C6×Dic3C6×Dic6 — C3×C122Q8
C3C2×C6 — C3×C122Q8
C1C2×C6C4×C12

Generators and relations for C3×C122Q8
 G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 306 in 151 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×6], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C32, Dic3 [×4], C12 [×12], C12 [×10], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], Dic6 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C3×Q8 [×4], C4⋊Q8, C3×Dic3 [×4], C3×C12 [×6], C62, C4⋊Dic3 [×4], C4×C12 [×2], C4×C12, C3×C4⋊C4 [×4], C2×Dic6 [×2], C6×Q8 [×2], C3×Dic6 [×4], C6×Dic3 [×4], C6×C12, C6×C12 [×2], C122Q8, C3×C4⋊Q8, C3×C4⋊Dic3 [×4], C122, C6×Dic6 [×2], C3×C122Q8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], Q8 [×4], C23, D6 [×3], C2×C6 [×7], C2×D4, C2×Q8 [×2], C3×S3, Dic6 [×4], D12 [×2], C3×D4 [×2], C3×Q8 [×4], C22×S3, C22×C6, C4⋊Q8, S3×C6 [×3], C2×Dic6 [×2], C2×D12, C6×D4, C6×Q8 [×2], C3×Dic6 [×4], C3×D12 [×2], S3×C2×C6, C122Q8, C3×C4⋊Q8, C6×Dic6 [×2], C6×D12, C3×C122Q8

Smallest permutation representation of C3×C122Q8
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(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 34 63 23)(2 35 64 24)(3 36 65 13)(4 25 66 14)(5 26 67 15)(6 27 68 16)(7 28 69 17)(8 29 70 18)(9 30 71 19)(10 31 72 20)(11 32 61 21)(12 33 62 22)(37 52 81 95)(38 53 82 96)(39 54 83 85)(40 55 84 86)(41 56 73 87)(42 57 74 88)(43 58 75 89)(44 59 76 90)(45 60 77 91)(46 49 78 92)(47 50 79 93)(48 51 80 94)
(1 84 63 40)(2 83 64 39)(3 82 65 38)(4 81 66 37)(5 80 67 48)(6 79 68 47)(7 78 69 46)(8 77 70 45)(9 76 71 44)(10 75 72 43)(11 74 61 42)(12 73 62 41)(13 96 36 53)(14 95 25 52)(15 94 26 51)(16 93 27 50)(17 92 28 49)(18 91 29 60)(19 90 30 59)(20 89 31 58)(21 88 32 57)(22 87 33 56)(23 86 34 55)(24 85 35 54)

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

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

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

90 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A···4F4G4H4I4J6A···6F6G···6O12A···12AV12AW···12BD
order1222333334···444446···66···612···1212···12
size1111112222···2121212121···12···22···212···12

90 irreducible representations

dim11111111222222222222
type++++++-+-+
imageC1C2C2C2C3C6C6C6S3D4Q8D6C3×S3Dic6D12C3×D4C3×Q8S3×C6C3×Dic6C3×D12
kernelC3×C122Q8C3×C4⋊Dic3C122C6×Dic6C122Q8C4⋊Dic3C4×C12C2×Dic6C4×C12C3×C12C3×C12C2×C12C42C12C12C12C12C2×C4C4C4
# reps141228241243284486168

Matrix representation of C3×C122Q8 in GL4(𝔽13) generated by

3000
0300
0030
0003
,
11000
0600
00110
0006
,
8000
0500
0010
0001
,
0100
12000
0001
0010
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[11,0,0,0,0,6,0,0,0,0,11,0,0,0,0,6],[8,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C122Q8 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes_2Q_8
% in TeX

G:=Group("C3xC12:2Q8");
// GroupNames label

G:=SmallGroup(288,640);
// by ID

G=gap.SmallGroup(288,640);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,590,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽