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G = C7×D12order 168 = 23·3·7

Direct product of C7 and D12

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C7×D12, C216D4, C845C2, C283S3, C121C14, D61C14, C14.15D6, C42.20C22, C4⋊(S3×C7), C31(C7×D4), (S3×C14)⋊4C2, C2.4(S3×C14), C6.3(C2×C14), SmallGroup(168,31)

Series: Derived Chief Lower central Upper central

C1C6 — C7×D12
C1C3C6C42S3×C14 — C7×D12
C3C6 — C7×D12
C1C14C28

Generators and relations for C7×D12
 G = < a,b,c | a7=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

6C2
6C2
3C22
3C22
2S3
2S3
6C14
6C14
3D4
3C2×C14
3C2×C14
2S3×C7
2S3×C7
3C7×D4

Smallest permutation representation of C7×D12
On 84 points
Generators in S84
(1 57 48 74 20 68 27)(2 58 37 75 21 69 28)(3 59 38 76 22 70 29)(4 60 39 77 23 71 30)(5 49 40 78 24 72 31)(6 50 41 79 13 61 32)(7 51 42 80 14 62 33)(8 52 43 81 15 63 34)(9 53 44 82 16 64 35)(10 54 45 83 17 65 36)(11 55 46 84 18 66 25)(12 56 47 73 19 67 26)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)(49 52)(50 51)(53 60)(54 59)(55 58)(56 57)(61 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 74)(75 84)(76 83)(77 82)(78 81)(79 80)

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

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

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

C7×D12 is a maximal subgroup of
C21⋊D8  C7⋊D24  C42.D4  D12.D7  D12⋊D7  D125D7  C28⋊D6  S3×C7×D4

63 conjugacy classes

class 1 2A2B2C 3  4  6 7A···7F12A12B14A···14F14G···14R21A···21F28A···28F42A···42F84A···84L
order12223467···7121214···1414···1421···2128···2842···4284···84
size11662221···1221···16···62···22···22···22···2

63 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C7C14C14S3D4D6D12S3×C7C7×D4S3×C14C7×D12
kernelC7×D12C84S3×C14D12C12D6C28C21C14C7C4C3C2C1
# reps1126612111266612

Matrix representation of C7×D12 in GL4(𝔽337) generated by

79000
07900
0010
0001
,
33633500
1100
000336
001336
,
33633500
0100
001336
000336
G:=sub<GL(4,GF(337))| [79,0,0,0,0,79,0,0,0,0,1,0,0,0,0,1],[336,1,0,0,335,1,0,0,0,0,0,1,0,0,336,336],[336,0,0,0,335,1,0,0,0,0,1,0,0,0,336,336] >;

C7×D12 in GAP, Magma, Sage, TeX

C_7\times D_{12}
% in TeX

G:=Group("C7xD12");
// GroupNames label

G:=SmallGroup(168,31);
// by ID

G=gap.SmallGroup(168,31);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-3,301,146,2804]);
// Polycyclic

G:=Group<a,b,c|a^7=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D12 in TeX

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