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G = C12×C3⋊C8order 288 = 25·32

Direct product of C12 and C3⋊C8

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C12×C3⋊C8, C122C24, C122.10C2, C31(C4×C24), (C3×C12)⋊7C8, C326(C4×C8), C6.1(C4×C12), C6.6(C2×C24), C4.18(S3×C12), (C6×C12).28C4, (C4×C12).26S3, (C4×C12).12C6, C42.6(C3×S3), C12.109(C4×S3), (C2×C12).15C12, C12.23(C2×C12), (C2×C12).450D6, C62.92(C2×C4), (C3×C6).11C42, C2.1(Dic3×C12), C6.17(C4×Dic3), (C2×C12).30Dic3, C22.7(C6×Dic3), (C6×C12).328C22, C2.1(C6×C3⋊C8), C6.17(C2×C3⋊C8), (C6×C3⋊C8).23C2, (C2×C3⋊C8).11C6, (C2×C4).87(S3×C6), (C3×C6).37(C2×C8), (C2×C6).34(C2×C12), (C2×C4).7(C3×Dic3), (C2×C12).117(C2×C6), (C3×C12).103(C2×C4), (C2×C6).56(C2×Dic3), SmallGroup(288,236)

Series: Derived Chief Lower central Upper central

C1C3 — C12×C3⋊C8
C1C3C6C2×C6C2×C12C6×C12C6×C3⋊C8 — C12×C3⋊C8
C3 — C12×C3⋊C8
C1C4×C12

Generators and relations for C12×C3⋊C8
 G = < a,b,c | a12=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 146 in 103 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×6], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×4], C2×C4, C2×C4 [×2], C32, C12 [×12], C12 [×6], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×3], C4×C8, C3×C12 [×6], C62, C2×C3⋊C8 [×2], C4×C12 [×2], C4×C12, C2×C24 [×2], C3×C3⋊C8 [×4], C6×C12, C6×C12 [×2], C4×C3⋊C8, C4×C24, C6×C3⋊C8 [×2], C122, C12×C3⋊C8
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C8 [×4], C2×C4 [×3], Dic3 [×2], C12 [×6], D6, C2×C6, C42, C2×C8 [×2], C3×S3, C3⋊C8 [×4], C24 [×4], C4×S3 [×2], C2×Dic3, C2×C12 [×3], C4×C8, C3×Dic3 [×2], S3×C6, C2×C3⋊C8 [×2], C4×Dic3, C4×C12, C2×C24 [×2], C3×C3⋊C8 [×4], S3×C12 [×2], C6×Dic3, C4×C3⋊C8, C4×C24, C6×C3⋊C8 [×2], Dic3×C12, C12×C3⋊C8

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

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

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

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

144 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A···4L6A···6F6G···6O8A···8P12A···12X12Y···12BH24A···24AF
order1222333334···46···66···68···812···1212···1224···24
size1111112221···11···12···23···31···12···23···3

144 irreducible representations

dim1111111111112222222222
type++++-+
imageC1C2C2C3C4C4C6C6C8C12C12C24S3Dic3D6C3×S3C3⋊C8C4×S3C3×Dic3S3×C6C3×C3⋊C8S3×C12
kernelC12×C3⋊C8C6×C3⋊C8C122C4×C3⋊C8C3×C3⋊C8C6×C12C2×C3⋊C8C4×C12C3×C12C3⋊C8C2×C12C12C4×C12C2×C12C2×C12C42C12C12C2×C4C2×C4C4C4
# reps12128442161683212128442168

Matrix representation of C12×C3⋊C8 in GL3(𝔽73) generated by

6400
0240
0024
,
100
0640
0468
,
5100
04327
01830
G:=sub<GL(3,GF(73))| [64,0,0,0,24,0,0,0,24],[1,0,0,0,64,46,0,0,8],[51,0,0,0,43,18,0,27,30] >;

C12×C3⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C12xC3:C8");
// GroupNames label

G:=SmallGroup(288,236);
// by ID

G=gap.SmallGroup(288,236);
# by ID

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

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

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