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## G = C2×D4.D5order 160 = 25·5

### Direct product of C2 and D4.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4.D5
 Chief series C1 — C5 — C10 — C20 — Dic10 — C2×Dic10 — C2×D4.D5
 Lower central C5 — C10 — C20 — C2×D4.D5
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4.D5
G = < a,b,c,d,e | a2=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 184 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, C10, C10 [×2], C10 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C2×SD16, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C2×C52C8, D4.D5 [×4], C2×Dic10, D4×C10, C2×D4.D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, C5⋊D4 [×2], C22×D5, D4.D5 [×2], C2×C5⋊D4, C2×D4.D5

Smallest permutation representation of C2×D4.D5
On 80 points
Generators in S80
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 46)(42 47)(43 48)(44 49)(45 50)(61 66)(62 67)(63 68)(64 69)(65 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 68 6 63)(2 67 7 62)(3 66 8 61)(4 70 9 65)(5 69 10 64)(11 73 16 78)(12 72 17 77)(13 71 18 76)(14 75 19 80)(15 74 20 79)(21 48 26 43)(22 47 27 42)(23 46 28 41)(24 50 29 45)(25 49 30 44)(31 53 36 58)(32 52 37 57)(33 51 38 56)(34 55 39 60)(35 54 40 59)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 1 1 4 4 2 2 20 20 2 2 10 10 10 10 2 ··· 2 4 ··· 4 4 4 4 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D4 D5 SD16 D10 D10 C5⋊D4 C5⋊D4 D4.D5 kernel C2×D4.D5 C2×C5⋊2C8 D4.D5 C2×Dic10 D4×C10 C20 C2×C10 C2×D4 C10 C2×C4 D4 C4 C22 C2 # reps 1 1 4 1 1 1 1 2 4 2 4 4 4 4

Matrix representation of C2×D4.D5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 22 18
,
 15 26 0 0 0 0 26 26 0 0 0 0 0 0 15 15 0 0 0 0 15 26 0 0 0 0 0 0 7 23 0 0 0 0 30 34

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,22,0,0,0,0,0,18],[15,26,0,0,0,0,26,26,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,7,30,0,0,0,0,23,34] >;

C2×D4.D5 in GAP, Magma, Sage, TeX

C_2\times D_4.D_5
% in TeX

G:=Group("C2xD4.D5");
// GroupNames label

G:=SmallGroup(160,154);
// by ID

G=gap.SmallGroup(160,154);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,579,159,69,4613]);
// Polycyclic

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

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