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G = C2×C40⋊C4order 320 = 26·5

Direct product of C2 and C40⋊C4

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

Aliases: C2×C40⋊C4, D10.10SD16, C88(C2×F5), (C2×C8)⋊7F5, (C2×C40)⋊7C4, C408(C2×C4), (C8×D5)⋊8C4, C10⋊(C4.Q8), D5⋊(C4.Q8), (C4×D5).82D4, C4.12(C4⋊F5), C20.19(C4⋊C4), (C4×D5).24Q8, D10.29(C2×D4), D5.1(C2×SD16), C4⋊F5.15C22, D10.27(C4⋊C4), C4.35(C22×F5), C20.75(C22×C4), Dic5.13(C2×Q8), (C2×Dic5).33Q8, (C22×D5).97D4, (C4×D5).75C23, (C8×D5).56C22, C22.22(C4⋊F5), Dic5.28(C4⋊C4), C5⋊(C2×C4.Q8), (D5×C2×C8).24C2, (C2×C52C8)⋊17C4, C2.14(C2×C4⋊F5), C52C834(C2×C4), C10.11(C2×C4⋊C4), (C2×C4⋊F5).13C2, (C4×D5).84(C2×C4), (C2×C4).136(C2×F5), (C2×C10).19(C4⋊C4), (C2×C20).126(C2×C4), (C2×C4×D5).394C22, SmallGroup(320,1057)

Series: Derived Chief Lower central Upper central

C1C20 — C2×C40⋊C4
C1C5C10D10C4×D5C4⋊F5C2×C4⋊F5 — C2×C40⋊C4
C5C10C20 — C2×C40⋊C4
C1C22C2×C4C2×C8

Generators and relations for C2×C40⋊C4
 G = < a,b,c | a2=b40=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 538 in 130 conjugacy classes, 60 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×4], C10, C10 [×2], C4⋊C4 [×6], C2×C8, C2×C8 [×5], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C4.Q8 [×4], C2×C4⋊C4 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×8], C22×D5, C2×C4.Q8, C8×D5 [×4], C2×C52C8, C2×C40, C4⋊F5 [×4], C4⋊F5 [×2], C2×C4×D5, C22×F5 [×2], C40⋊C4 [×4], D5×C2×C8, C2×C4⋊F5 [×2], C2×C40⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, F5, C4.Q8 [×4], C2×C4⋊C4, C2×SD16 [×2], C2×F5 [×3], C2×C4.Q8, C4⋊F5 [×2], C22×F5, C40⋊C4 [×2], C2×C4⋊F5, C2×C40⋊C4

Smallest permutation representation of C2×C40⋊C4
On 80 points
Generators in S80
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)
(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)
(1 26 21 6)(2 13 30 9)(3 40 39 12)(4 27 8 15)(5 14 17 18)(7 28 35 24)(10 29 22 33)(11 16 31 36)(19 32 23 20)(25 34 37 38)(41 46 61 66)(42 73 70 69)(43 60 79 72)(44 47 48 75)(45 74 57 78)(49 62 53 50)(51 76 71 56)(52 63 80 59)(54 77 58 65)(55 64 67 68)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4L 5 8A8B8C8D8E8F8G8H10A10B10C20A20B20C20D40A···40H
order1222222244444···45888888881010102020202040···40
size1111555522101020···20422221010101044444444···4

44 irreducible representations

dim111111122222444444
type+++++--++++
imageC1C2C2C2C4C4C4D4Q8Q8D4SD16F5C2×F5C2×F5C4⋊F5C4⋊F5C40⋊C4
kernelC2×C40⋊C4C40⋊C4D5×C2×C8C2×C4⋊F5C8×D5C2×C52C8C2×C40C4×D5C4×D5C2×Dic5C22×D5D10C2×C8C8C2×C4C4C22C2
# reps141242211118121228

Matrix representation of C2×C40⋊C4 in GL8(𝔽41)

400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
030000000
2630000000
0035310000
001660000
00002727034
0000734347
00003402727
0000147140
,
1724000000
2924000000
0028330000
0011130000
0000701414
0000141407
00002734270
0000347734

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,26,0,0,0,0,0,0,30,30,0,0,0,0,0,0,0,0,35,16,0,0,0,0,0,0,31,6,0,0,0,0,0,0,0,0,27,7,34,14,0,0,0,0,27,34,0,7,0,0,0,0,0,34,27,14,0,0,0,0,34,7,27,0],[17,29,0,0,0,0,0,0,24,24,0,0,0,0,0,0,0,0,28,11,0,0,0,0,0,0,33,13,0,0,0,0,0,0,0,0,7,14,27,34,0,0,0,0,0,14,34,7,0,0,0,0,14,0,27,7,0,0,0,0,14,7,0,34] >;

C2×C40⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{40}\rtimes C_4
% in TeX

G:=Group("C2xC40:C4");
// GroupNames label

G:=SmallGroup(320,1057);
// by ID

G=gap.SmallGroup(320,1057);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,1684,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^2=b^40=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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