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## G = C2×D5⋊C8order 160 = 25·5

### Direct product of C2 and D5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×D5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C2×D5⋊C8
 Lower central C5 — C2×D5⋊C8
 Upper central C1 — C2×C4

Generators and relations for C2×D5⋊C8
G = < a,b,c,d | a2=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Subgroups: 196 in 76 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C22×C8, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, D5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×D5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, C22×C8, C2×F5, D5⋊C8, C22×F5, C2×D5⋊C8

Smallest permutation representation of C2×D5⋊C8
On 80 points
Generators in S80
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 77)(26 78)(27 79)(28 80)(29 73)(30 74)(31 75)(32 76)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(1 21 75 33 57)(2 34 22 58 76)(3 59 35 77 23)(4 78 60 24 36)(5 17 79 37 61)(6 38 18 62 80)(7 63 39 73 19)(8 74 64 20 40)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(1 55)(2 32)(3 13)(4 42)(5 51)(6 28)(7 9)(8 46)(10 74)(11 33)(12 22)(14 78)(15 37)(16 18)(17 43)(19 71)(20 30)(21 47)(23 67)(24 26)(25 59)(27 79)(29 63)(31 75)(34 56)(35 41)(36 68)(38 52)(39 45)(40 72)(44 62)(48 58)(49 77)(50 60)(53 73)(54 64)(57 65)(61 69)(66 76)(70 80)
(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)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5 8A ··· 8P 10A 10B 10C 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 8 ··· 8 10 10 10 20 20 20 20 size 1 1 1 1 5 5 5 5 1 1 1 1 5 5 5 5 4 5 ··· 5 4 4 4 4 4 4 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 F5 C2×F5 C2×F5 D5⋊C8 kernel C2×D5⋊C8 D5⋊C8 C2×C5⋊C8 C2×C4×D5 C4×D5 C2×C20 C22×D5 D10 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 16 1 2 1 4

Matrix representation of C2×D5⋊C8 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 40 0 1 0 0 40 0 0 1 0 40 0 0 0 1 40
,
 1 0 0 0 0 0 0 0 40 1 0 0 40 0 1 0 40 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 0 11 30 12 0 0 23 30 0 11 0 11 0 30 23 0 0 12 30 11

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[1,0,0,0,0,0,0,0,40,0,0,0,40,0,0,0,40,0,0,0,0,1,1,1,1],[40,0,0,0,0,0,11,23,11,0,0,30,30,0,12,0,12,0,30,30,0,0,11,23,11] >;

C2×D5⋊C8 in GAP, Magma, Sage, TeX

C_2\times D_5\rtimes C_8
% in TeX

G:=Group("C2xD5:C8");
// GroupNames label

G:=SmallGroup(160,200);
// by ID

G=gap.SmallGroup(160,200);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,69,2309,599]);
// Polycyclic

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

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