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## G = C2×D10.D4order 320 = 26·5

### Direct product of C2 and D10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×D10.D4
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C22⋊F5 — C2×C22⋊F5 — C2×D10.D4
 Lower central C5 — C10 — C2×C10 — C2×D10.D4
 Upper central C1 — C22 — C23 — C22×C4

Generators and relations for C2×D10.D4
G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=b3, dcd-1=b7c, ece-1=b2c, ede-1=b4cd-1 >

Subgroups: 1194 in 210 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×6], C22 [×3], C22 [×22], C5, C2×C4 [×2], C2×C4 [×10], D4 [×8], C23, C23 [×14], D5 [×6], C10, C10 [×2], C10 [×2], C22⋊C4 [×6], C22×C4, C22×C4 [×2], C2×D4 [×8], C24 [×2], C20 [×2], F5 [×4], D10 [×4], D10 [×16], C2×C10 [×3], C2×C10 [×2], C23⋊C4 [×4], C2×C22⋊C4 [×2], C22×D4, D20 [×8], C2×C20 [×2], C2×C20 [×2], C2×F5 [×8], C22×D5 [×2], C22×D5 [×6], C22×D5 [×6], C22×C10, C2×C23⋊C4, C22⋊F5 [×4], C22⋊F5 [×2], C2×D20 [×4], C2×D20 [×4], C22×C20, C22×F5 [×2], C23×D5 [×2], D10.D4 [×4], C2×C22⋊F5 [×2], C22×D20, C2×D10.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C23⋊C4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C23⋊C4, C22⋊F5 [×2], C22×F5, D10.D4 [×2], C2×C22⋊F5, C2×D10.D4

Smallest permutation representation of C2×D10.D4
On 80 points
Generators in S80
(1 47)(2 48)(3 49)(4 50)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 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 10)(2 9)(3 8)(4 7)(5 6)(11 12)(13 20)(14 19)(15 18)(16 17)(21 27)(22 26)(23 25)(28 30)(31 37)(32 36)(33 35)(38 40)(41 42)(43 50)(44 49)(45 48)(46 47)(51 52)(53 60)(54 59)(55 58)(56 57)(61 67)(62 66)(63 65)(68 70)(71 77)(72 76)(73 75)(78 80)
(1 22 12 32)(2 29 11 35)(3 26 20 38)(4 23 19 31)(5 30 18 34)(6 27 17 37)(7 24 16 40)(8 21 15 33)(9 28 14 36)(10 25 13 39)(41 70 58 74)(42 67 57 77)(43 64 56 80)(44 61 55 73)(45 68 54 76)(46 65 53 79)(47 62 52 72)(48 69 51 75)(49 66 60 78)(50 63 59 71)
(2 8 10 4)(3 5 9 7)(11 20 13 14)(12 17)(15 18 19 16)(21 35 28 34)(22 32 27 37)(23 39 26 40)(24 36 25 33)(29 31 30 38)(41 45 43 49)(44 46 50 48)(51 60 53 54)(52 57)(55 58 59 56)(61 75 68 74)(62 72 67 77)(63 79 66 80)(64 76 65 73)(69 71 70 78)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 F5 C23⋊C4 C2×F5 C2×F5 C22⋊F5 D10.D4 kernel C2×D10.D4 D10.D4 C2×C22⋊F5 C22×D20 C2×D20 C22×C20 C23×D5 C22×D5 C22×C4 C10 C2×C4 C23 C22 C2 # reps 1 4 2 1 4 2 2 4 1 2 2 1 4 8

Matrix representation of C2×D10.D4 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 1 1 1 0 0 0 0 40 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 0 40 40 40 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 1 1 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 37 33 40 34 0 0 0 0 7 1 8 4 0 0 0 0 7 3 40 6 0 0 0 0 37 3 38 4
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1,0,0,0,0,0,0,40,1,0],[0,0,1,40,0,0,0,0,0,1,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,40,0,1,0,0,0,0,40,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,37,7,7,37,0,0,0,0,33,1,3,3,0,0,0,0,40,8,40,38,0,0,0,0,34,4,6,4],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40] >;

C2×D10.D4 in GAP, Magma, Sage, TeX

C_2\times D_{10}.D_4
% in TeX

G:=Group("C2xD10.D4");
// GroupNames label

G:=SmallGroup(320,1082);
// by ID

G=gap.SmallGroup(320,1082);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,297,1684,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=b^-1*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=b^3,d*c*d^-1=b^7*c,e*c*e^-1=b^2*c,e*d*e^-1=b^4*c*d^-1>;
// generators/relations

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