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## G = C5×C4⋊D4order 160 = 25·5

### Direct product of C5 and C4⋊D4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C4⋊D4
G = < a,b,c,d | a5=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×2], D4 [×6], C23, C23 [×2], C10 [×3], C10 [×4], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C4⋊D4, C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×C10, C22×C10 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, D4×C10, D4×C10 [×2], C5×C4⋊D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×4], C23, C10 [×7], C2×D4 [×2], C4○D4, C2×C10 [×7], C4⋊D4, C5×D4 [×4], C22×C10, D4×C10 [×2], C5×C4○D4, C5×C4⋊D4

Smallest permutation representation of C5×C4⋊D4
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 66 26 56)(2 67 27 57)(3 68 28 58)(4 69 29 59)(5 70 30 60)(6 41 16 51)(7 42 17 52)(8 43 18 53)(9 44 19 54)(10 45 20 55)(11 46 76 36)(12 47 77 37)(13 48 78 38)(14 49 79 39)(15 50 80 40)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)
(1 36 21 41)(2 37 22 42)(3 38 23 43)(4 39 24 44)(5 40 25 45)(6 66 76 71)(7 67 77 72)(8 68 78 73)(9 69 79 74)(10 70 80 75)(11 61 16 56)(12 62 17 57)(13 63 18 58)(14 64 19 59)(15 65 20 60)(26 46 31 51)(27 47 32 52)(28 48 33 53)(29 49 34 54)(30 50 35 55)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 D4 D4 C4○D4 C5×D4 C5×D4 C5×C4○D4 kernel C5×C4⋊D4 C5×C22⋊C4 C5×C4⋊C4 C22×C20 D4×C10 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C20 C2×C10 C10 C4 C22 C2 # reps 1 2 1 1 3 4 8 4 4 12 2 2 2 8 8 8

Matrix representation of C5×C4⋊D4 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 40 0
,
 40 1 0 0 39 1 0 0 0 0 40 0 0 0 0 1
,
 1 40 0 0 0 40 0 0 0 0 1 0 0 0 0 40
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[40,39,0,0,1,1,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,40,40,0,0,0,0,1,0,0,0,0,40] >;

C5×C4⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes D_4
% in TeX

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

G:=SmallGroup(160,182);
// by ID

G=gap.SmallGroup(160,182);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,247,1514]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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