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G = C3×C12⋊C8order 288 = 25·32

Direct product of C3 and C12⋊C8

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C12⋊C8, C121C24, C12.91D12, C122.6C2, C12.34Dic6, C123(C3⋊C8), (C3×C12)⋊6C8, C328(C4⋊C8), C6.7(C2×C24), C12.7(C3×Q8), (C2×C12).9C12, (C4×C12).22S3, (C6×C12).16C4, (C4×C12).10C6, C4.16(C3×D12), C12.32(C3×D4), C42.2(C3×S3), (C3×C12).25Q8, C4.7(C3×Dic6), (C2×C12).452D6, (C3×C12).134D4, C62.94(C2×C4), C6.5(C3×M4(2)), C6.17(C4⋊Dic3), (C2×C12).27Dic3, (C3×C6).19M4(2), C22.9(C6×Dic3), (C6×C12).330C22, C6.12(C4.Dic3), C4⋊(C3×C3⋊C8), C31(C3×C4⋊C8), C2.3(C6×C3⋊C8), C6.1(C3×C4⋊C4), (C2×C3⋊C8).8C6, (C6×C3⋊C8).5C2, C6.18(C2×C3⋊C8), (C3×C6).38(C2×C8), (C2×C4).89(S3×C6), C2.1(C3×C4⋊Dic3), (C3×C6).27(C4⋊C4), (C2×C6).36(C2×C12), (C2×C4).4(C3×Dic3), (C2×C12).119(C2×C6), C2.2(C3×C4.Dic3), (C2×C6).58(C2×Dic3), SmallGroup(288,238)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C12⋊C8
C1C3C6C2×C6C2×C12C6×C12C6×C3⋊C8 — C3×C12⋊C8
C3C6 — C3×C12⋊C8
C1C2×C12C4×C12

Generators and relations for C3×C12⋊C8
 G = < a,b,c | a3=b12=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 146 in 91 conjugacy classes, 58 normal (46 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C4, C22, C6 [×6], C6 [×3], C8 [×2], C2×C4 [×3], C32, C12 [×4], C12 [×4], C12 [×8], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×C12 [×6], C2×C12 [×3], C4⋊C8, C3×C12 [×2], C3×C12 [×2], C3×C12, C62, C2×C3⋊C8 [×2], C4×C12 [×2], C4×C12, C2×C24 [×2], C3×C3⋊C8 [×2], C6×C12 [×3], C12⋊C8, C3×C4⋊C8, C6×C3⋊C8 [×2], C122, C3×C12⋊C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C2×C8, M4(2), C3×S3, C3⋊C8 [×2], C24 [×2], Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C4⋊C8, C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C3×M4(2), C3×C3⋊C8 [×2], C3×Dic6, C3×D12, C6×Dic3, C12⋊C8, C3×C4⋊C8, C6×C3⋊C8, C3×C4.Dic3, C3×C4⋊Dic3, C3×C12⋊C8

Smallest permutation representation of C3×C12⋊C8
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(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 26 74 19 93 58 41 68)(2 25 75 18 94 57 42 67)(3 36 76 17 95 56 43 66)(4 35 77 16 96 55 44 65)(5 34 78 15 85 54 45 64)(6 33 79 14 86 53 46 63)(7 32 80 13 87 52 47 62)(8 31 81 24 88 51 48 61)(9 30 82 23 89 50 37 72)(10 29 83 22 90 49 38 71)(11 28 84 21 91 60 39 70)(12 27 73 20 92 59 40 69)

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

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

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

108 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F4G4H6A···6F6G···6O8A···8H12A···12H12I···12AZ24A···24P
order122233333444444446···66···68···812···1212···1224···24
size111111222111122221···12···26···61···12···26···6

108 irreducible representations

dim111111111122222222222222222222
type+++++--+-+
imageC1C2C2C3C4C6C6C8C12C24S3D4Q8Dic3D6M4(2)C3×S3C3⋊C8Dic6D12C3×D4C3×Q8C3×Dic3S3×C6C4.Dic3C3×M4(2)C3×C3⋊C8C3×Dic6C3×D12C3×C4.Dic3
kernelC3×C12⋊C8C6×C3⋊C8C122C12⋊C8C6×C12C2×C3⋊C8C4×C12C3×C12C2×C12C12C4×C12C3×C12C3×C12C2×C12C2×C12C3×C6C42C12C12C12C12C12C2×C4C2×C4C6C6C4C4C4C2
# reps1212442881611121224222242448448

Matrix representation of C3×C12⋊C8 in GL4(𝔽73) generated by

8000
0800
0080
0008
,
65000
0900
00356
00049
,
0100
46000
004638
004327
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[65,0,0,0,0,9,0,0,0,0,3,0,0,0,56,49],[0,46,0,0,1,0,0,0,0,0,46,43,0,0,38,27] >;

C3×C12⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes C_8
% in TeX

G:=Group("C3xC12:C8");
// GroupNames label

G:=SmallGroup(288,238);
// by ID

G=gap.SmallGroup(288,238);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,176,136,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^12=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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