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G = Dic3xC2xC12order 288 = 25·32

Direct product of C2xC12 and Dic3

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

Aliases: Dic3xC2xC12, C62.190C23, C6:(C4xC12), C12:8(C2xC12), (C6xC12):16C4, (C2xC12):7C12, (C3xC6):3C42, C32:7(C2xC42), (C2xC12).463D6, C23.38(S3xC6), C62.79(C2xC4), C6.22(C22xC12), (C22xC12).26C6, (C22xC12).45S3, C22.15(S3xC12), (C22xC6).170D6, (C6xC12).351C22, (C2xC62).93C22, (C22xDic3).8C6, C22.13(C6xDic3), C6.42(C22xDic3), (C6xDic3).167C22, C3:2(C2xC4xC12), C2.3(S3xC2xC12), (C2xC6xC12).21C2, C6.116(S3xC2xC4), (C3xC12):24(C2xC4), C2.2(Dic3xC2xC6), (C2xC6).85(C4xS3), C22.19(S3xC2xC6), (C2xC4).101(S3xC6), (C2xC6).20(C2xC12), (Dic3xC2xC6).14C2, (C2xC12).131(C2xC6), (C2xC6).45(C22xC6), (C22xC4).14(C3xS3), (C22xC6).57(C2xC6), (C3xC6).88(C22xC4), (C2xC6).62(C2xDic3), (C2xC6).323(C22xS3), (C2xDic3).49(C2xC6), SmallGroup(288,693)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3xC2xC12
C1C3C6C2xC6C62C6xDic3Dic3xC2xC6 — Dic3xC2xC12
C3 — Dic3xC2xC12
C1C22xC12

Generators and relations for Dic3xC2xC12
 G = < a,b,c,d | a2=b12=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 378 in 243 conjugacy classes, 162 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2xC4, C2xC4, C23, C32, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22xC4, C22xC4, C3xC6, C3xC6, C2xDic3, C2xC12, C2xC12, C22xC6, C22xC6, C2xC42, C3xDic3, C3xC12, C62, C62, C4xDic3, C4xC12, C22xDic3, C22xC12, C22xC12, C6xDic3, C6xC12, C2xC62, C2xC4xDic3, C2xC4xC12, Dic3xC12, Dic3xC2xC6, C2xC6xC12, Dic3xC2xC12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C23, Dic3, C12, D6, C2xC6, C42, C22xC4, C3xS3, C4xS3, C2xDic3, C2xC12, C22xS3, C22xC6, C2xC42, C3xDic3, S3xC6, C4xDic3, C4xC12, S3xC2xC4, C22xDic3, C22xC12, S3xC12, C6xDic3, S3xC2xC6, C2xC4xDic3, C2xC4xC12, Dic3xC12, S3xC2xC12, Dic3xC2xC6, Dic3xC2xC12

Smallest permutation representation of Dic3xC2xC12
On 96 points
Generators in S96
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 85)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)
(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 37 5 41 9 45)(2 38 6 42 10 46)(3 39 7 43 11 47)(4 40 8 44 12 48)(13 56 21 52 17 60)(14 57 22 53 18 49)(15 58 23 54 19 50)(16 59 24 55 20 51)(25 82 33 78 29 74)(26 83 34 79 30 75)(27 84 35 80 31 76)(28 73 36 81 32 77)(61 87 65 91 69 95)(62 88 66 92 70 96)(63 89 67 93 71 85)(64 90 68 94 72 86)
(1 77 41 36)(2 78 42 25)(3 79 43 26)(4 80 44 27)(5 81 45 28)(6 82 46 29)(7 83 47 30)(8 84 48 31)(9 73 37 32)(10 74 38 33)(11 75 39 34)(12 76 40 35)(13 65 52 95)(14 66 53 96)(15 67 54 85)(16 68 55 86)(17 69 56 87)(18 70 57 88)(19 71 58 89)(20 72 59 90)(21 61 60 91)(22 62 49 92)(23 63 50 93)(24 64 51 94)

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

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

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

144 conjugacy classes

class 1 2A···2G3A3B3C3D3E4A···4H4I···4X6A···6N6O···6AI12A···12P12Q···12AN12AO···12BT
order12···2333334···44···46···66···612···1212···1212···12
size11···1112221···13···31···12···21···12···23···3

144 irreducible representations

dim1111111111112222222222
type+++++-++
imageC1C2C2C2C3C4C4C6C6C6C12C12S3Dic3D6D6C3xS3C4xS3C3xDic3S3xC6S3xC6S3xC12
kernelDic3xC2xC12Dic3xC12Dic3xC2xC6C2xC6xC12C2xC4xDic3C6xDic3C6xC12C4xDic3C22xDic3C22xC12C2xDic3C2xC12C22xC12C2xC12C2xC12C22xC6C22xC4C2xC6C2xC4C2xC4C23C22
# reps14212168842321614212884216

Matrix representation of Dic3xC2xC12 in GL5(F13)

120000
01000
001200
00010
00001
,
70000
02000
00400
000100
000010
,
120000
01000
00100
00090
00003
,
50000
012000
001200
000012
000120

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,10],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;

Dic3xC2xC12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_{12}
% in TeX

G:=Group("Dic3xC2xC12");
// GroupNames label

G:=SmallGroup(288,693);
// by ID

G=gap.SmallGroup(288,693);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,268,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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