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## G = C2×C40⋊C4order 320 = 26·5

### Direct product of C2 and C40⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C40⋊C4
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — C2×C4⋊F5 — C2×C40⋊C4
 Lower central C5 — C10 — C20 — C2×C40⋊C4
 Upper central C1 — C22 — C2×C4 — C2×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 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4L 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 8 8 8 8 8 8 8 8 10 10 10 20 20 20 20 40 ··· 40 size 1 1 1 1 5 5 5 5 2 2 10 10 20 ··· 20 4 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - - + + + + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 Q8 D4 SD16 F5 C2×F5 C2×F5 C4⋊F5 C4⋊F5 C40⋊C4 kernel C2×C40⋊C4 C40⋊C4 D5×C2×C8 C2×C4⋊F5 C8×D5 C2×C5⋊2C8 C2×C40 C4×D5 C4×D5 C2×Dic5 C22×D5 D10 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 1 1 1 1 8 1 2 1 2 2 8

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

 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 30 0 0 0 0 0 0 26 30 0 0 0 0 0 0 0 0 35 31 0 0 0 0 0 0 16 6 0 0 0 0 0 0 0 0 27 27 0 34 0 0 0 0 7 34 34 7 0 0 0 0 34 0 27 27 0 0 0 0 14 7 14 0
,
 17 24 0 0 0 0 0 0 29 24 0 0 0 0 0 0 0 0 28 33 0 0 0 0 0 0 11 13 0 0 0 0 0 0 0 0 7 0 14 14 0 0 0 0 14 14 0 7 0 0 0 0 27 34 27 0 0 0 0 0 34 7 7 34

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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