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## G = C2×D4⋊8D10order 320 = 26·5

### Direct product of C2 and D4⋊8D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4⋊8D10
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — C2×D4×D5 — C2×D4⋊8D10
 Lower central C5 — C10 — C2×D4⋊8D10
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C2×D48D10
G = < a,b,c,d,e | a2=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ce=ec, ede=d-1 >

Subgroups: 3182 in 898 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×8], C4 [×4], C22, C22 [×6], C22 [×54], C5, C2×C4, C2×C4 [×15], C2×C4 [×26], D4 [×12], D4 [×60], Q8 [×4], Q8 [×4], C23 [×3], C23 [×42], D5 [×12], C10, C10 [×2], C10 [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×D4 [×87], C2×Q8, C2×Q8, C4○D4 [×8], C4○D4 [×40], C24 [×6], Dic5 [×4], C20 [×8], D10 [×12], D10 [×36], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×D4 [×9], C2×C4○D4, C2×C4○D4 [×5], 2+ 1+4 [×16], Dic10 [×4], C4×D5 [×24], D20 [×36], C2×Dic5 [×2], C5⋊D4 [×24], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×D5 [×30], C22×D5 [×12], C22×C10 [×3], C2×2+ 1+4, C2×Dic10, C2×C4×D5 [×6], C2×D20 [×33], C4○D20 [×24], D4×D5 [×48], Q82D5 [×16], C2×C5⋊D4 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C23×D5 [×6], C22×D20 [×3], C2×C4○D20 [×3], C2×D4×D5 [×6], C2×Q82D5 [×2], D48D10 [×16], C10×C4○D4, C2×D48D10
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2+ 1+4 [×2], C25, C22×D5 [×35], C2×2+ 1+4, C23×D5 [×15], D48D10 [×2], D5×C24, C2×D48D10

Smallest permutation representation of C2×D48D10
On 80 points
Generators in S80
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 60)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)
(1 20 60 22)(2 11 51 23)(3 12 52 24)(4 13 53 25)(5 14 54 26)(6 15 55 27)(7 16 56 28)(8 17 57 29)(9 18 58 30)(10 19 59 21)(31 72 70 45)(32 73 61 46)(33 74 62 47)(34 75 63 48)(35 76 64 49)(36 77 65 50)(37 78 66 41)(38 79 67 42)(39 80 68 43)(40 71 69 44)
(1 70)(2 32)(3 62)(4 34)(5 64)(6 36)(7 66)(8 38)(9 68)(10 40)(11 46)(12 74)(13 48)(14 76)(15 50)(16 78)(17 42)(18 80)(19 44)(20 72)(21 71)(22 45)(23 73)(24 47)(25 75)(26 49)(27 77)(28 41)(29 79)(30 43)(31 60)(33 52)(35 54)(37 56)(39 58)(51 61)(53 63)(55 65)(57 67)(59 69)
(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 5)(2 4)(6 10)(7 9)(11 25)(12 24)(13 23)(14 22)(15 21)(16 30)(17 29)(18 28)(19 27)(20 26)(31 35)(32 34)(36 40)(37 39)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 53)(54 60)(55 59)(56 58)(61 63)(64 70)(65 69)(66 68)

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2U 4A ··· 4H 4I 4J 4K 4L 5A 5B 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 10 ··· 10 2 ··· 2 10 10 10 10 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 D10 2+ 1+4 D4⋊8D10 kernel C2×D4⋊8D10 C22×D20 C2×C4○D20 C2×D4×D5 C2×Q8⋊2D5 D4⋊8D10 C10×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C10 C2 # reps 1 3 3 6 2 16 1 2 6 6 2 16 2 8

Matrix representation of C2×D48D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 1 0 0 0 0 0 0 1 0 0
,
 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 40 0 0 0 0 0 0 40
,
 7 34 0 0 0 0 7 40 0 0 0 0 0 0 0 0 2 16 0 0 0 0 25 16 0 0 39 25 0 0 0 0 16 25 0 0
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 6 0 0 0 0 0 1 0 0 0 0 0 0 1 35 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,7,0,0,0,0,34,40,0,0,0,0,0,0,0,0,39,16,0,0,0,0,25,25,0,0,2,25,0,0,0,0,16,16,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,0,1,0,0,0,0,0,35,40] >;

C2×D48D10 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_8D_{10}
% in TeX

G:=Group("C2xD4:8D10");
// GroupNames label

G:=SmallGroup(320,1619);
// by ID

G=gap.SmallGroup(320,1619);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^10=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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