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## G = C5×D4⋊C4order 160 = 25·5

### Direct product of C5 and D4⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×D4⋊C4
G = < a,b,c,d | a5=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

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

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 4 4 2 2 4 4 1 1 1 1 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D4 D4 D8 SD16 C5×D4 C5×D4 C5×D8 C5×SD16 kernel C5×D4⋊C4 C5×C4⋊C4 C2×C40 D4×C10 C5×D4 D4⋊C4 C4⋊C4 C2×C8 C2×D4 D4 C20 C2×C10 C10 C10 C4 C22 C2 C2 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2 2 4 4 8 8

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

 18 0 0 0 0 18 0 0 0 0 10 0 0 0 0 10
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 1 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 40 0 0 0 0 0 29 12 0 0 12 12
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[0,40,0,0,1,0,0,0,0,0,29,12,0,0,12,12] >;

C5×D4⋊C4 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes C_4
% in TeX

G:=Group("C5xD4:C4");
// GroupNames label

G:=SmallGroup(160,52);
// by ID

G=gap.SmallGroup(160,52);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,2403,1209,117]);
// Polycyclic

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

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