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G = C3xDic3:5D4order 288 = 25·32

Direct product of C3 and Dic3:5D4

direct product, metabelian, supersoluble, monomial

Aliases: C3xDic3:5D4, D12:5C12, C62.184C23, C12:8(C4xS3), C4:1(S3xC12), C3:3(D4xC12), C12:2(C2xC12), D6:C4:12C6, (C3xD12):9C4, D6:3(C2xC12), C6.24(C6xD4), C32:20(C4xD4), Dic3:5(C3xD4), (C4xDic3):3C6, (C2xD12).7C6, C6.183(S3xD4), (C3xDic3):20D4, (C6xD12).12C2, (C2xC12).271D6, (Dic3xC12):13C2, C6.11(C22xC12), (C6xC12).249C22, C6.58(Q8:3S3), (C6xDic3).127C22, (C3xC4:C4):4C6, C4:C4:8(C3xS3), C2.4(C3xS3xD4), (S3xC2xC4):12C6, (C3xC4:C4):17S3, (S3xC2xC12):26C2, C2.13(S3xC2xC12), C6.110(S3xC2xC4), (S3xC6):16(C2xC4), (C3xC12):10(C2xC4), (C3xD6:C4):28C2, (C32xC4:C4):5C2, (C2xC4).43(S3xC6), C6.33(C3xC4oD4), C22.18(S3xC2xC6), (C2xC12).58(C2xC6), (C3xC6).212(C2xD4), (S3xC2xC6).93C22, C2.2(C3xQ8:3S3), (C3xC6).82(C22xC4), (C2xC6).39(C22xC6), (C3xC6).155(C4oD4), (C22xS3).20(C2xC6), (C2xC6).317(C22xS3), (C2xDic3).48(C2xC6), SmallGroup(288,664)

Series: Derived Chief Lower central Upper central

C1C6 — C3xDic3:5D4
C1C3C6C2xC6C62S3xC2xC6C6xD12 — C3xDic3:5D4
C3C6 — C3xDic3:5D4
C1C2xC6C3xC4:C4

Generators and relations for C3xDic3:5D4
 G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 466 in 201 conjugacy classes, 86 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3xC6, C4xS3, D12, C2xDic3, C2xC12, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C4xD4, C3xDic3, C3xDic3, C3xC12, C3xC12, S3xC6, S3xC6, C62, C4xDic3, D6:C4, C4xC12, C3xC22:C4, C3xC4:C4, C3xC4:C4, S3xC2xC4, C2xD12, C22xC12, C6xD4, S3xC12, C3xD12, C6xDic3, C6xC12, C6xC12, S3xC2xC6, Dic3:5D4, D4xC12, Dic3xC12, C3xD6:C4, C32xC4:C4, S3xC2xC12, C6xD12, C3xDic3:5D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, D4, C23, C12, D6, C2xC6, C22xC4, C2xD4, C4oD4, C3xS3, C4xS3, C2xC12, C3xD4, C22xS3, C22xC6, C4xD4, S3xC6, S3xC2xC4, S3xD4, Q8:3S3, C22xC12, C6xD4, C3xC4oD4, S3xC12, S3xC2xC6, Dic3:5D4, D4xC12, S3xC2xC12, C3xS3xD4, C3xQ8:3S3, C3xDic3:5D4

Smallest permutation representation of C3xDic3:5D4
On 96 points
Generators in S96
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 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 69 4 72)(2 68 5 71)(3 67 6 70)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 77 16 74)(14 76 17 73)(15 75 18 78)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 62 28 65)(26 61 29 64)(27 66 30 63)(37 91 40 94)(38 96 41 93)(39 95 42 92)(43 87 46 90)(44 86 47 89)(45 85 48 88)(49 83 52 80)(50 82 53 79)(51 81 54 84)
(1 41 17 36)(2 42 18 31)(3 37 13 32)(4 38 14 33)(5 39 15 34)(6 40 16 35)(7 72 96 76)(8 67 91 77)(9 68 92 78)(10 69 93 73)(11 70 94 74)(12 71 95 75)(19 49 30 43)(20 50 25 44)(21 51 26 45)(22 52 27 46)(23 53 28 47)(24 54 29 48)(55 79 65 89)(56 80 66 90)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)
(1 88)(2 87)(3 86)(4 85)(5 90)(6 89)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 82)(14 81)(15 80)(16 79)(17 84)(18 83)(25 94)(26 93)(27 92)(28 91)(29 96)(30 95)(31 59)(32 58)(33 57)(34 56)(35 55)(36 60)(37 62)(38 61)(39 66)(40 65)(41 64)(42 63)(43 71)(44 70)(45 69)(46 68)(47 67)(48 72)(49 75)(50 74)(51 73)(52 78)(53 77)(54 76)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A···4F4G4H4I4J4K4L6A···6F6G···6O6P···6W12A···12L12M···12T12U···12AL12AM12AN12AO12AP
order12222222333334···44444446···66···66···612···1212···1212···1212121212
size11116666112222···23333661···12···26···62···23···34···46666

90 irreducible representations

dim1111111111111122222222224444
type+++++++++++
imageC1C2C2C2C2C2C3C4C6C6C6C6C6C12S3D4D6C4oD4C3xS3C3xD4C4xS3S3xC6C3xC4oD4S3xC12S3xD4Q8:3S3C3xS3xD4C3xQ8:3S3
kernelC3xDic3:5D4Dic3xC12C3xD6:C4C32xC4:C4S3xC2xC12C6xD12Dic3:5D4C3xD12C4xDic3D6:C4C3xC4:C4S3xC2xC4C2xD12D12C3xC4:C4C3xDic3C2xC12C3xC6C4:C4Dic3C12C2xC4C6C4C6C6C2C2
# reps11212128242421612322446481122

Matrix representation of C3xDic3:5D4 in GL5(F13)

30000
03000
00300
00010
00001
,
120000
03000
00900
00010
00001
,
50000
00100
01000
000120
000012
,
10000
01000
00100
000111
000112
,
120000
00100
01000
000111
000012

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,11,12],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,12] >;

C3xDic3:5D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\rtimes_5D_4
% in TeX

G:=Group("C3xDic3:5D4");
// GroupNames label

G:=SmallGroup(288,664);
// by ID

G=gap.SmallGroup(288,664);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,555,142,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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