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## G = C4×C5⋊D4order 160 = 25·5

### Direct product of C4 and C5⋊D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C5⋊D4
G = < a,b,c,d | a4=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 264 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×C5⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C5⋊D4, C22×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4

Smallest permutation representation of C4×C5⋊D4
On 80 points
Generators in S80
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(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)
(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 6 46)(2 45 7 50)(3 44 8 49)(4 43 9 48)(5 42 10 47)(11 51 16 56)(12 55 17 60)(13 54 18 59)(14 53 19 58)(15 52 20 57)(21 61 26 66)(22 65 27 70)(23 64 28 69)(24 63 29 68)(25 62 30 67)(31 71 36 76)(32 75 37 80)(33 74 38 79)(34 73 39 78)(35 72 40 77)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G ··· 4L 5A 5B 10A ··· 10N 20A ··· 20P order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 10 10 1 1 1 1 2 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

52 irreducible representations

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

Matrix representation of C4×C5⋊D4 in GL3(𝔽41) generated by

 9 0 0 0 32 0 0 0 32
,
 1 0 0 0 6 40 0 1 0
,
 40 0 0 0 23 6 0 21 18
,
 1 0 0 0 40 0 0 35 1
G:=sub<GL(3,GF(41))| [9,0,0,0,32,0,0,0,32],[1,0,0,0,6,1,0,40,0],[40,0,0,0,23,21,0,6,18],[1,0,0,0,40,35,0,0,1] >;

C4×C5⋊D4 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes D_4
% in TeX

G:=Group("C4xC5:D4");
// GroupNames label

G:=SmallGroup(160,149);
// by ID

G=gap.SmallGroup(160,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^5=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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