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G = Dic3×C21order 252 = 22·32·7

Direct product of C21 and Dic3

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

Aliases: Dic3×C21, C3⋊C84, C6.C42, C217C12, C42.8S3, C322C28, C42.13C6, (C3×C21)⋊6C4, C2.(S3×C21), C6.4(S3×C7), C14.4(C3×S3), (C3×C6).1C14, (C3×C42).4C2, SmallGroup(252,21)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C21
C1C3C6C42C3×C42 — Dic3×C21
C3 — Dic3×C21
C1C42

Generators and relations for Dic3×C21
 G = < a,b,c | a21=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

2C3
3C4
2C6
2C21
3C12
3C28
2C42
3C84

Smallest permutation representation of Dic3×C21
On 84 points
Generators in S84
(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 79 8 65 15 72)(2 80 9 66 16 73)(3 81 10 67 17 74)(4 82 11 68 18 75)(5 83 12 69 19 76)(6 84 13 70 20 77)(7 64 14 71 21 78)(22 63 36 56 29 49)(23 43 37 57 30 50)(24 44 38 58 31 51)(25 45 39 59 32 52)(26 46 40 60 33 53)(27 47 41 61 34 54)(28 48 42 62 35 55)
(1 44 65 31)(2 45 66 32)(3 46 67 33)(4 47 68 34)(5 48 69 35)(6 49 70 36)(7 50 71 37)(8 51 72 38)(9 52 73 39)(10 53 74 40)(11 54 75 41)(12 55 76 42)(13 56 77 22)(14 57 78 23)(15 58 79 24)(16 59 80 25)(17 60 81 26)(18 61 82 27)(19 62 83 28)(20 63 84 29)(21 43 64 30)

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

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

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

126 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E7A···7F12A12B12C12D14A···14F21A···21L21M···21AD28A···28L42A···42L42M···42AD84A···84X
order123333344666667···71212121214···1421···2121···2128···2842···4242···4284···84
size111122233112221···133331···11···12···23···31···12···23···3

126 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C3C4C6C7C12C14C21C28C42C84S3Dic3C3×S3C3×Dic3S3×C7C7×Dic3S3×C21Dic3×C21
kernelDic3×C21C3×C42C7×Dic3C3×C21C42C3×Dic3C21C3×C6Dic3C32C6C3C42C21C14C7C6C3C2C1
# reps11222646121212241122661212

Matrix representation of Dic3×C21 in GL2(𝔽43) generated by

380
038
,
370
07
,
02
210
G:=sub<GL(2,GF(43))| [38,0,0,38],[37,0,0,7],[0,21,2,0] >;

Dic3×C21 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{21}
% in TeX

G:=Group("Dic3xC21");
// GroupNames label

G:=SmallGroup(252,21);
// by ID

G=gap.SmallGroup(252,21);
# by ID

G:=PCGroup([5,-2,-3,-7,-2,-3,210,4204]);
// Polycyclic

G:=Group<a,b,c|a^21=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C21 in TeX

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