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## G = C2×D10⋊C4order 160 = 25·5

### Direct product of C2 and D10⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D10⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D10⋊C4
 Lower central C5 — C10 — C2×D10⋊C4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×D10⋊C4
G = < a,b,c,d | a2=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >

Subgroups: 456 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C22, C22 [×6], C22 [×16], C5, C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×4], C22×C4, C22×C4, C24, Dic5 [×2], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×6], C22×D5 [×4], C22×C10, D10⋊C4 [×4], C22×Dic5, C22×C20, C23×D5, C2×D10⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4

Smallest permutation representation of C2×D10⋊C4
On 80 points
Generators in S80
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 76)(22 77)(23 78)(24 79)(25 80)(26 71)(27 72)(28 73)(29 74)(30 75)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 60)(8 59)(9 58)(10 57)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 80)(29 79)(30 78)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 70)(39 69)(40 68)
(1 37 17 27)(2 38 18 28)(3 39 19 29)(4 40 20 30)(5 31 11 21)(6 32 12 22)(7 33 13 23)(8 34 14 24)(9 35 15 25)(10 36 16 26)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)

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

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

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

52 conjugacy classes

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

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 kernel C2×D10⋊C4 D10⋊C4 C22×Dic5 C22×C20 C23×D5 C22×D5 C2×C10 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 8 4 2 4 2 8 8 8

Matrix representation of C2×D10⋊C4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 6 34 0 0 0 6 0 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40
,
 32 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,6,6,0,0,0,34,0,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,40],[32,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×D10⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes C_4
% in TeX

G:=Group("C2xD10:C4");
// GroupNames label

G:=SmallGroup(160,148);
// by ID

G=gap.SmallGroup(160,148);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations

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