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G = C3xDic21order 252 = 22·32·7

Direct product of C3 and Dic21

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

Aliases: C3xDic21, C21:5C12, C42.9C6, C42.4S3, C6.4D21, C21:2Dic3, C32:2Dic7, C6.(C3xD7), (C3xC21):4C4, C3:(C3xDic7), C2.(C3xD21), (C3xC6).1D7, C7:3(C3xDic3), C14.3(C3xS3), (C3xC42).2C2, SmallGroup(252,22)

Series: Derived Chief Lower central Upper central

C1C21 — C3xDic21
C1C7C21C42C3xC42 — C3xDic21
C21 — C3xDic21
C1C6

Generators and relations for C3xDic21
 G = < a,b,c | a3=b42=1, c2=b21, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 88 in 28 conjugacy classes, 18 normal (all characteristic)
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, D7, C3xS3, Dic7, C3xDic3, C3xD7, D21, C3xDic7, Dic21, C3xD21, C3xDic21
2C3
21C4
2C6
2C21
7Dic3
21C12
3Dic7
2C42
7C3xDic3
3C3xDic7

Smallest permutation representation of C3xDic21
On 84 points
Generators in S84
(1 15 29)(2 16 30)(3 17 31)(4 18 32)(5 19 33)(6 20 34)(7 21 35)(8 22 36)(9 23 37)(10 24 38)(11 25 39)(12 26 40)(13 27 41)(14 28 42)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)
(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)
(1 57 22 78)(2 56 23 77)(3 55 24 76)(4 54 25 75)(5 53 26 74)(6 52 27 73)(7 51 28 72)(8 50 29 71)(9 49 30 70)(10 48 31 69)(11 47 32 68)(12 46 33 67)(13 45 34 66)(14 44 35 65)(15 43 36 64)(16 84 37 63)(17 83 38 62)(18 82 39 61)(19 81 40 60)(20 80 41 59)(21 79 42 58)

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

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

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

72 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E7A7B7C12A12B12C12D14A14B14C21A···21X42A···42X
order123333344666667771212121214141421···2142···42
size1111222212111222222212121212222···22···2

72 irreducible representations

dim111111222222222222
type+++-+-+-
imageC1C2C3C4C6C12S3Dic3D7C3xS3Dic7C3xDic3C3xD7D21C3xDic7Dic21C3xD21C3xDic21
kernelC3xDic21C3xC42Dic21C3xC21C42C21C42C21C3xC6C14C32C7C6C6C3C3C2C1
# reps11222411323266661212

Matrix representation of C3xDic21 in GL2(F43) generated by

60
06
,
280
020
,
042
10
G:=sub<GL(2,GF(43))| [6,0,0,6],[28,0,0,20],[0,1,42,0] >;

C3xDic21 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{21}
% in TeX

G:=Group("C3xDic21");
// GroupNames label

G:=SmallGroup(252,22);
// by ID

G=gap.SmallGroup(252,22);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-7,30,483,5404]);
// Polycyclic

G:=Group<a,b,c|a^3=b^42=1,c^2=b^21,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3xDic21 in TeX

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