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G = C2×C41⋊C4order 328 = 23·41

Direct product of C2 and C41⋊C4

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

Aliases: C2×C41⋊C4, C82⋊C4, D41⋊C4, D82.C2, D41.C22, C41⋊(C2×C4), SmallGroup(328,13)

Series: Derived Chief Lower central Upper central

C1C41 — C2×C41⋊C4
C1C41D41C41⋊C4 — C2×C41⋊C4
C41 — C2×C41⋊C4
C1C2

Generators and relations for C2×C41⋊C4
 G = < a,b,c | a2=b41=c4=1, ab=ba, ac=ca, cbc-1=b9 >

41C2
41C2
41C4
41C22
41C4
41C2×C4

Character table of C2×C41⋊C4

 class 12A2B2C4A4B4C4D41A41B41C41D41E41F41G41H41I41J82A82B82C82D82E82F82G82H82I82J
 size 1141414141414144444444444444444444
ρ11111111111111111111111111111    trivial
ρ21-11-11-11-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31-11-1-11-111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-1-111111111111111111111    linear of order 2
ρ51-1-11i-i-ii1111111111-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ611-1-1ii-i-i11111111111111111111    linear of order 4
ρ711-1-1-i-iii11111111111111111111    linear of order 4
ρ81-1-11-iii-i1111111111-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ944000000ζ4125412141204116ζ413341314110418ζ413841274114413ζ413541284113416ζ4129412641154112ζ413941234118412ζ4130412441174111ζ41374136415414ζ4140413241941ζ413441224119417ζ4140413241941ζ413441224119417ζ4125412141204116ζ413341314110418ζ413841274114413ζ413541284113416ζ4129412641154112ζ413941234118412ζ4130412441174111ζ41374136415414    orthogonal lifted from C41⋊C4
ρ104-4000000ζ413941234118412ζ4140413241941ζ4129412641154112ζ4130412441174111ζ413441224119417ζ413341314110418ζ413841274114413ζ4125412141204116ζ41374136415414ζ413541284113416413741364154144135412841134164139412341184124140413241941412941264115411241304124411741114134412241194174133413141104184138412741144134125412141204116    orthogonal faithful
ρ1144000000ζ413441224119417ζ4130412441174111ζ4140413241941ζ413941234118412ζ41374136415414ζ413541284113416ζ413341314110418ζ4129412641154112ζ413841274114413ζ4125412141204116ζ413841274114413ζ4125412141204116ζ413441224119417ζ4130412441174111ζ4140413241941ζ413941234118412ζ41374136415414ζ413541284113416ζ413341314110418ζ4129412641154112    orthogonal lifted from C41⋊C4
ρ124-4000000ζ413441224119417ζ4130412441174111ζ4140413241941ζ413941234118412ζ41374136415414ζ413541284113416ζ413341314110418ζ4129412641154112ζ413841274114413ζ4125412141204116413841274114413412541214120411641344122411941741304124411741114140413241941413941234118412413741364154144135412841134164133413141104184129412641154112    orthogonal faithful
ρ134-4000000ζ4140413241941ζ4125412141204116ζ413541284113416ζ4129412641154112ζ4130412441174111ζ41374136415414ζ413441224119417ζ413341314110418ζ413941234118412ζ413841274114413413941234118412413841274114413414041324194141254121412041164135412841134164129412641154112413041244117411141374136415414413441224119417413341314110418    orthogonal faithful
ρ144-4000000ζ413541284113416ζ413841274114413ζ41374136415414ζ413341314110418ζ4125412141204116ζ4130412441174111ζ4140413241941ζ413441224119417ζ4129412641154112ζ413941234118412412941264115411241394123411841241354128411341641384127411441341374136415414413341314110418412541214120411641304124411741114140413241941413441224119417    orthogonal faithful
ρ154-4000000ζ4129412641154112ζ413541284113416ζ413341314110418ζ4125412141204116ζ4140413241941ζ413441224119417ζ413941234118412ζ413841274114413ζ4130412441174111ζ41374136415414413041244117411141374136415414412941264115411241354128411341641334131411041841254121412041164140413241941413441224119417413941234118412413841274114413    orthogonal faithful
ρ164-4000000ζ4130412441174111ζ4129412641154112ζ4125412141204116ζ4140413241941ζ413941234118412ζ413841274114413ζ41374136415414ζ413541284113416ζ413441224119417ζ413341314110418413441224119417413341314110418413041244117411141294126411541124125412141204116414041324194141394123411841241384127411441341374136415414413541284113416    orthogonal faithful
ρ1744000000ζ4140413241941ζ4125412141204116ζ413541284113416ζ4129412641154112ζ4130412441174111ζ41374136415414ζ413441224119417ζ413341314110418ζ413941234118412ζ413841274114413ζ413941234118412ζ413841274114413ζ4140413241941ζ4125412141204116ζ413541284113416ζ4129412641154112ζ4130412441174111ζ41374136415414ζ413441224119417ζ413341314110418    orthogonal lifted from C41⋊C4
ρ184-4000000ζ41374136415414ζ413941234118412ζ4130412441174111ζ413441224119417ζ413841274114413ζ4125412141204116ζ413541284113416ζ4140413241941ζ413341314110418ζ4129412641154112413341314110418412941264115411241374136415414413941234118412413041244117411141344122411941741384127411441341254121412041164135412841134164140413241941    orthogonal faithful
ρ1944000000ζ413541284113416ζ413841274114413ζ41374136415414ζ413341314110418ζ4125412141204116ζ4130412441174111ζ4140413241941ζ413441224119417ζ4129412641154112ζ413941234118412ζ4129412641154112ζ413941234118412ζ413541284113416ζ413841274114413ζ41374136415414ζ413341314110418ζ4125412141204116ζ4130412441174111ζ4140413241941ζ413441224119417    orthogonal lifted from C41⋊C4
ρ2044000000ζ413941234118412ζ4140413241941ζ4129412641154112ζ4130412441174111ζ413441224119417ζ413341314110418ζ413841274114413ζ4125412141204116ζ41374136415414ζ413541284113416ζ41374136415414ζ413541284113416ζ413941234118412ζ4140413241941ζ4129412641154112ζ4130412441174111ζ413441224119417ζ413341314110418ζ413841274114413ζ4125412141204116    orthogonal lifted from C41⋊C4
ρ2144000000ζ413841274114413ζ413441224119417ζ413941234118412ζ41374136415414ζ413341314110418ζ4129412641154112ζ4125412141204116ζ4130412441174111ζ413541284113416ζ4140413241941ζ413541284113416ζ4140413241941ζ413841274114413ζ413441224119417ζ413941234118412ζ41374136415414ζ413341314110418ζ4129412641154112ζ4125412141204116ζ4130412441174111    orthogonal lifted from C41⋊C4
ρ224-4000000ζ413841274114413ζ413441224119417ζ413941234118412ζ41374136415414ζ413341314110418ζ4129412641154112ζ4125412141204116ζ4130412441174111ζ413541284113416ζ4140413241941413541284113416414041324194141384127411441341344122411941741394123411841241374136415414413341314110418412941264115411241254121412041164130412441174111    orthogonal faithful
ρ234-4000000ζ413341314110418ζ41374136415414ζ413441224119417ζ413841274114413ζ413541284113416ζ4140413241941ζ4129412641154112ζ413941234118412ζ4125412141204116ζ4130412441174111412541214120411641304124411741114133413141104184137413641541441344122411941741384127411441341354128411341641404132419414129412641154112413941234118412    orthogonal faithful
ρ244-4000000ζ4125412141204116ζ413341314110418ζ413841274114413ζ413541284113416ζ4129412641154112ζ413941234118412ζ4130412441174111ζ41374136415414ζ4140413241941ζ413441224119417414041324194141344122411941741254121412041164133413141104184138412741144134135412841134164129412641154112413941234118412413041244117411141374136415414    orthogonal faithful
ρ2544000000ζ413341314110418ζ41374136415414ζ413441224119417ζ413841274114413ζ413541284113416ζ4140413241941ζ4129412641154112ζ413941234118412ζ4125412141204116ζ4130412441174111ζ4125412141204116ζ4130412441174111ζ413341314110418ζ41374136415414ζ413441224119417ζ413841274114413ζ413541284113416ζ4140413241941ζ4129412641154112ζ413941234118412    orthogonal lifted from C41⋊C4
ρ2644000000ζ41374136415414ζ413941234118412ζ4130412441174111ζ413441224119417ζ413841274114413ζ4125412141204116ζ413541284113416ζ4140413241941ζ413341314110418ζ4129412641154112ζ413341314110418ζ4129412641154112ζ41374136415414ζ413941234118412ζ4130412441174111ζ413441224119417ζ413841274114413ζ4125412141204116ζ413541284113416ζ4140413241941    orthogonal lifted from C41⋊C4
ρ2744000000ζ4130412441174111ζ4129412641154112ζ4125412141204116ζ4140413241941ζ413941234118412ζ413841274114413ζ41374136415414ζ413541284113416ζ413441224119417ζ413341314110418ζ413441224119417ζ413341314110418ζ4130412441174111ζ4129412641154112ζ4125412141204116ζ4140413241941ζ413941234118412ζ413841274114413ζ41374136415414ζ413541284113416    orthogonal lifted from C41⋊C4
ρ2844000000ζ4129412641154112ζ413541284113416ζ413341314110418ζ4125412141204116ζ4140413241941ζ413441224119417ζ413941234118412ζ413841274114413ζ4130412441174111ζ41374136415414ζ4130412441174111ζ41374136415414ζ4129412641154112ζ413541284113416ζ413341314110418ζ4125412141204116ζ4140413241941ζ413441224119417ζ413941234118412ζ413841274114413    orthogonal lifted from C41⋊C4

Smallest permutation representation of C2×C41⋊C4
On 82 points
Generators in S82
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)
(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)
(2 33 41 10)(3 24 40 19)(4 15 39 28)(5 6 38 37)(7 29 36 14)(8 20 35 23)(9 11 34 32)(12 25 31 18)(13 16 30 27)(17 21 26 22)(43 74 82 51)(44 65 81 60)(45 56 80 69)(46 47 79 78)(48 70 77 55)(49 61 76 64)(50 52 75 73)(53 66 72 59)(54 57 71 68)(58 62 67 63)

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

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

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

Matrix representation of C2×C41⋊C4 in GL5(𝔽821)

8200000
01000
00100
00010
00001
,
10000
020660980820
0466741340201
050637361420
034022469194
,
2950000
0725146284161
0685489288706
073535225599
0320772668203

G:=sub<GL(5,GF(821))| [820,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,206,466,506,340,0,609,741,373,224,0,80,340,61,691,0,820,201,420,94],[295,0,0,0,0,0,725,685,735,320,0,146,489,35,772,0,284,288,225,668,0,161,706,599,203] >;

C2×C41⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{41}\rtimes C_4
% in TeX

G:=Group("C2xC41:C4");
// GroupNames label

G:=SmallGroup(328,13);
// by ID

G=gap.SmallGroup(328,13);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,16,4099,1291]);
// Polycyclic

G:=Group<a,b,c|a^2=b^41=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

Export

Subgroup lattice of C2×C41⋊C4 in TeX
Character table of C2×C41⋊C4 in TeX

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