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G = C2xC6xD12order 288 = 25·32

Direct product of C2xC6 and D12

direct product, metabelian, supersoluble, monomial

Aliases: C2xC6xD12, C62:20D4, C62.267C23, C6:1(C6xD4), (C2xC12):31D6, (S3xC23):6C6, (S3xC6):8C23, (C3xC12):8C23, C12:9(C22xS3), C12:2(C22xC6), D6:1(C22xC6), C6.3(C23xC6), C32:9(C22xD4), (C6xC12):34C22, (C22xC12):11C6, (C22xC12):16S3, C23.45(S3xC6), (C3xC6).40C24, C6.71(S3xC23), (C22xC6).176D6, (C2xC62).117C22, C4:2(S3xC2xC6), C3:1(D4xC2xC6), (C2xC6xC12):13C2, (C2xC4):9(S3xC6), (C3xC6):8(C2xD4), (C2xC6):9(C3xD4), (S3xC22xC6):8C2, (C2xC12):12(C2xC6), C2.4(S3xC22xC6), (S3xC2xC6):19C22, (C22xC4):9(C3xS3), C22.30(S3xC2xC6), (C22xS3):6(C2xC6), (C2xC6).69(C22xC6), (C22xC6).70(C2xC6), (C2xC6).347(C22xS3), SmallGroup(288,990)

Series: Derived Chief Lower central Upper central

C1C6 — C2xC6xD12
C1C3C6C3xC6S3xC6S3xC2xC6S3xC22xC6 — C2xC6xD12
C3C6 — C2xC6xD12
C1C22xC6C22xC12

Generators and relations for C2xC6xD12
 G = < a,b,c,d | a2=b6=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1210 in 499 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, D4, C23, C23, C32, C12, C12, D6, D6, C2xC6, C2xC6, C22xC4, C2xD4, C24, C3xS3, C3xC6, C3xC6, D12, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22xC6, C22xD4, C3xC12, S3xC6, S3xC6, C62, C2xD12, C22xC12, C22xC12, C6xD4, S3xC23, C23xC6, C3xD12, C6xC12, S3xC2xC6, S3xC2xC6, C2xC62, C22xD12, D4xC2xC6, C6xD12, C2xC6xC12, S3xC22xC6, C2xC6xD12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C24, C3xS3, D12, C3xD4, C22xS3, C22xC6, C22xD4, S3xC6, C2xD12, C6xD4, S3xC23, C23xC6, C3xD12, S3xC2xC6, C22xD12, D4xC2xC6, C6xD12, S3xC22xC6, C2xC6xD12

Smallest permutation representation of C2xC6xD12
On 96 points
Generators in S96
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 95)(14 96)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 73)(46 74)(47 75)(48 76)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 24 5 16 9 20)(2 13 6 17 10 21)(3 14 7 18 11 22)(4 15 8 19 12 23)(25 88 29 92 33 96)(26 89 30 93 34 85)(27 90 31 94 35 86)(28 91 32 95 36 87)(37 60 45 56 41 52)(38 49 46 57 42 53)(39 50 47 58 43 54)(40 51 48 59 44 55)(61 81 69 77 65 73)(62 82 70 78 66 74)(63 83 71 79 67 75)(64 84 72 80 68 76)
(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 75)(2 74)(3 73)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 62)(14 61)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 41)(26 40)(27 39)(28 38)(29 37)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 96)(57 95)(58 94)(59 93)(60 92)

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

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

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

108 conjugacy classes

class 1 2A···2G2H···2O3A3B3C3D3E4A4B4C4D6A···6N6O···6AI6AJ···6AY12A···12AF
order12···22···23333344446···66···66···612···12
size11···16···61122222221···12···26···62···2

108 irreducible representations

dim111111112222222222
type+++++++++
imageC1C2C2C2C3C6C6C6S3D4D6D6C3xS3D12C3xD4S3xC6S3xC6C3xD12
kernelC2xC6xD12C6xD12C2xC6xC12S3xC22xC6C22xD12C2xD12C22xC12S3xC23C22xC12C62C2xC12C22xC6C22xC4C2xC6C2xC6C2xC4C23C22
# reps1121222424146128812216

Matrix representation of C2xC6xD12 in GL5(F13)

120000
012000
001200
000120
000012
,
100000
03000
00300
00090
00009
,
120000
04000
031000
00020
00087
,
10000
03600
031000
00094
00064

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[10,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9],[12,0,0,0,0,0,4,3,0,0,0,0,10,0,0,0,0,0,2,8,0,0,0,0,7],[1,0,0,0,0,0,3,3,0,0,0,6,10,0,0,0,0,0,9,6,0,0,0,4,4] >;

C2xC6xD12 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_{12}
% in TeX

G:=Group("C2xC6xD12");
// GroupNames label

G:=SmallGroup(288,990);
// by ID

G=gap.SmallGroup(288,990);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,1571,192,9414]);
// Polycyclic

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

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