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## G = C2×D8⋊D5order 320 = 26·5

### Direct product of C2 and D8⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D8⋊D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4×D5 — C2×D8⋊D5
 Lower central C5 — C10 — C20 — C2×D8⋊D5
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for C2×D8⋊D5
G = < a,b,c,d,e | a2=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1310 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×24], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×4], D4 [×13], Q8 [×3], C23 [×12], D5 [×4], C10, C10 [×2], C10 [×4], C2×C8, C2×C8, M4(2) [×4], D8 [×4], D8 [×4], SD16 [×8], C22×C4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×6], C24, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×8], C2×M4(2), C2×D8, C2×D8, C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C52C8 [×2], C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×D5, C22×D5 [×9], C22×C10 [×2], C2×C8⋊C22, C8⋊D5 [×4], C40⋊C2 [×4], C2×C52C8, D4⋊D5 [×4], D4.D5 [×4], C2×C40, C5×D8 [×4], C2×Dic10, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], D42D5 [×4], D42D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], D4×C10 [×2], C23×D5, C2×C8⋊D5, C2×C40⋊C2, D8⋊D5 [×8], C2×D4⋊D5, C2×D4.D5, C10×D8, C2×D4×D5, C2×D42D5, C2×D8⋊D5
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C22×D5 [×7], C2×C8⋊C22, D4×D5 [×2], C23×D5, D8⋊D5 [×2], C2×D4×D5, C2×D8⋊D5

Smallest permutation representation of C2×D8⋊D5
On 80 points
Generators in S80
(1 68)(2 69)(3 70)(4 71)(5 72)(6 65)(7 66)(8 67)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 80)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(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 7)(2 6)(3 5)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(26 32)(27 31)(28 30)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)(49 51)(52 56)(53 55)(58 64)(59 63)(60 62)(65 69)(66 68)(70 72)(73 79)(74 78)(75 77)
(1 77 38 15 58)(2 78 39 16 59)(3 79 40 9 60)(4 80 33 10 61)(5 73 34 11 62)(6 74 35 12 63)(7 75 36 13 64)(8 76 37 14 57)(17 43 52 69 31)(18 44 53 70 32)(19 45 54 71 25)(20 46 55 72 26)(21 47 56 65 27)(22 48 49 66 28)(23 41 50 67 29)(24 42 51 68 30)
(1 55)(2 52)(3 49)(4 54)(5 51)(6 56)(7 53)(8 50)(9 28)(10 25)(11 30)(12 27)(13 32)(14 29)(15 26)(16 31)(17 39)(18 36)(19 33)(20 38)(21 35)(22 40)(23 37)(24 34)(41 76)(42 73)(43 78)(44 75)(45 80)(46 77)(47 74)(48 79)(57 67)(58 72)(59 69)(60 66)(61 71)(62 68)(63 65)(64 70)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 4 4 10 10 20 20 2 2 10 10 20 20 2 2 4 4 20 20 2 ··· 2 8 ··· 8 4 4 4 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 C8⋊C22 D4×D5 D4×D5 D8⋊D5 kernel C2×D8⋊D5 C2×C8⋊D5 C2×C40⋊C2 D8⋊D5 C2×D4⋊D5 C2×D4.D5 C10×D8 C2×D4×D5 C2×D4⋊2D5 C4×D5 C2×Dic5 C22×D5 C2×D8 C2×C8 D8 C2×D4 C10 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 2 1 1 2 2 8 4 2 2 2 8

Matrix representation of C2×D8⋊D5 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
,
 40 0 2 0 0 0 0 0 0 40 0 2 0 0 0 0 40 0 1 0 0 0 0 0 0 40 0 1 0 0 0 0 0 0 0 0 0 0 10 31 0 0 0 0 0 0 0 39 0 0 0 0 4 21 19 20 0 0 0 0 0 21 21 22
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 1 0 0 0 0 0 0 40 0 1 0 0 0 0 0 0 0 0 40 0 21 21 0 0 0 0 0 40 40 38 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 40 0 0 0 0 0 0 8 7 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 6
,
 35 1 0 0 0 0 0 0 6 6 0 0 0 0 0 0 0 0 35 1 0 0 0 0 0 0 6 6 0 0 0 0 0 0 0 0 0 35 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 6 35

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],[40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,2,0,1,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,21,21,0,0,0,0,10,0,19,21,0,0,0,0,31,39,20,22],[40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,21,40,1,0,0,0,0,0,21,38,0,1],[6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,8,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6],[35,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,35,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35] >;

C2×D8⋊D5 in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes D_5
% in TeX

G:=Group("C2xD8:D5");
// GroupNames label

G:=SmallGroup(320,1427);
// by ID

G=gap.SmallGroup(320,1427);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,185,438,235,102,12550]);
// Polycyclic

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

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