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## G = C5×C4⋊1D4order 160 = 25·5

### Direct product of C5 and C4⋊1D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C4⋊1D4
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C4⋊1D4
 Lower central C1 — C22 — C5×C4⋊1D4
 Upper central C1 — C2×C10 — C5×C4⋊1D4

Generators and relations for C5×C41D4
G = < a,b,c,d | a5=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×12], C5, C2×C4 [×3], D4 [×12], C23 [×4], C10 [×3], C10 [×4], C42, C2×D4 [×6], C20 [×6], C2×C10, C2×C10 [×12], C41D4, C2×C20 [×3], C5×D4 [×12], C22×C10 [×4], C4×C20, D4×C10 [×6], C5×C41D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], C23, C10 [×7], C2×D4 [×3], C2×C10 [×7], C41D4, C5×D4 [×6], C22×C10, D4×C10 [×3], C5×C41D4

Smallest permutation representation of C5×C41D4
On 80 points
Generators in S80
(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 11 31 6)(2 12 32 7)(3 13 33 8)(4 14 34 9)(5 15 35 10)(16 26 76 21)(17 27 77 22)(18 28 78 23)(19 29 79 24)(20 30 80 25)(36 71 51 56)(37 72 52 57)(38 73 53 58)(39 74 54 59)(40 75 55 60)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)
(1 41 21 36)(2 42 22 37)(3 43 23 38)(4 44 24 39)(5 45 25 40)(6 61 76 56)(7 62 77 57)(8 63 78 58)(9 64 79 59)(10 65 80 60)(11 66 16 71)(12 67 17 72)(13 68 18 73)(14 69 19 74)(15 70 20 75)(26 51 31 46)(27 52 32 47)(28 53 33 48)(29 54 34 49)(30 55 35 50)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 71)(27 72)(28 73)(29 74)(30 75)(31 66)(32 67)(33 68)(34 69)(35 70)(36 76)(37 77)(38 78)(39 79)(40 80)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AB 20A ··· 20X order 1 2 2 2 2 2 2 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 4 4 4 4 2 ··· 2 1 1 1 1 1 ··· 1 4 ··· 4 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C5 C10 C10 D4 C5×D4 kernel C5×C4⋊1D4 C4×C20 D4×C10 C4⋊1D4 C42 C2×D4 C20 C4 # reps 1 1 6 4 4 24 6 24

Matrix representation of C5×C41D4 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 32 4 0 0 0 9 0 0 0 0 40 39 0 0 1 1
,
 40 0 0 0 0 40 0 0 0 0 40 39 0 0 1 1
,
 9 37 0 0 20 32 0 0 0 0 1 0 0 0 40 40
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[32,0,0,0,4,9,0,0,0,0,40,1,0,0,39,1],[40,0,0,0,0,40,0,0,0,0,40,1,0,0,39,1],[9,20,0,0,37,32,0,0,0,0,1,40,0,0,0,40] >;

C5×C41D4 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes_1D_4
% in TeX

G:=Group("C5xC4:1D4");
// GroupNames label

G:=SmallGroup(160,188);
// by ID

G=gap.SmallGroup(160,188);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,247,1514,374]);
// Polycyclic

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

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