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## G = C5×D8⋊2C4order 320 = 26·5

### Direct product of C5 and D8⋊2C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×D8⋊2C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C4.Q8 — C5×D8⋊2C4
 Lower central C1 — C2 — C4 — C8 — C5×D8⋊2C4
 Upper central C1 — C10 — C2×C20 — C2×C40 — C5×D8⋊2C4

Generators and relations for C5×D82C4
G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C4.Q8, M5(2), C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, D82C4, C80, C5×C4⋊C4, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C4.Q8, C5×M5(2), C5×C4○D8, C5×D82C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, D8, SD16, C20, C2×C10, D4⋊C4, C2×C20, C5×D4, D82C4, C5×C22⋊C4, C5×D8, C5×SD16, C5×D4⋊C4, C5×D82C4

Smallest permutation representation of C5×D82C4
On 80 points
Generators in S80
(1 39 31 23 15)(2 40 32 24 16)(3 33 25 17 9)(4 34 26 18 10)(5 35 27 19 11)(6 36 28 20 12)(7 37 29 21 13)(8 38 30 22 14)(41 73 65 57 49)(42 74 66 58 50)(43 75 67 59 51)(44 76 68 60 52)(45 77 69 61 53)(46 78 70 62 54)(47 79 71 63 55)(48 80 72 64 56)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 56)(16 55)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 64)(24 63)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 72)(32 71)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 80)(40 79)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 42 45 46)(43 48 47 44)(49 50 53 54)(51 56 55 52)(57 58 61 62)(59 64 63 60)(65 66 69 70)(67 72 71 68)(73 74 77 78)(75 80 79 76)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 8C 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 16A 16B 16C 16D 20A ··· 20H 20I ··· 20T 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 4 4 4 4 4 5 5 5 5 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 16 16 16 16 20 ··· 20 20 ··· 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 8 2 2 8 8 8 1 1 1 1 2 2 4 1 1 1 1 2 2 2 2 8 8 8 8 4 4 4 4 2 ··· 2 8 ··· 8 2 ··· 2 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 D4 SD16 D8 C5×D4 C5×D4 C5×SD16 C5×D8 D8⋊2C4 C5×D8⋊2C4 kernel C5×D8⋊2C4 C5×C4.Q8 C5×M5(2) C5×C4○D8 C5×D8 C5×Q16 D8⋊2C4 C4.Q8 M5(2) C4○D8 D8 Q16 C40 C2×C20 C20 C2×C10 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 2 2 4 4 8 8 2 8

Matrix representation of C5×D82C4 in GL4(𝔽241) generated by

 87 0 0 0 0 87 0 0 0 0 87 0 0 0 0 87
,
 0 19 43 32 203 38 166 11 0 0 222 222 0 0 19 222
,
 64 11 158 121 0 0 19 222 203 0 166 11 203 38 166 11
,
 240 0 11 75 239 1 199 64 0 0 222 222 0 0 222 19
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[0,203,0,0,19,38,0,0,43,166,222,19,32,11,222,222],[64,0,203,203,11,0,0,38,158,19,166,166,121,222,11,11],[240,239,0,0,0,1,0,0,11,199,222,222,75,64,222,19] >;

C5×D82C4 in GAP, Magma, Sage, TeX

C_5\times D_8\rtimes_2C_4
% in TeX

G:=Group("C5xD8:2C4");
// GroupNames label

G:=SmallGroup(320,165);
// by ID

G=gap.SmallGroup(320,165);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,3650,136,3511,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations

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