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## G = C2×D4.D10order 320 = 26·5

### Direct product of C2 and D4.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4.D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×D4.D10
 Lower central C5 — C10 — C20 — C2×D4.D10
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for C2×D4.D10
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d9 >

Subgroups: 958 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×4], D4 [×13], Q8 [×3], C23, C23 [×11], D5 [×2], C10, C10 [×2], C10 [×6], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×5], C2×Q8, C4○D4 [×6], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×18], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C52C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×10], C2×C8⋊C22, C2×C52C8 [×2], C4.Dic5 [×4], D4⋊D5 [×8], D4.D5 [×8], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, D4×C10 [×6], D4×C10 [×3], C23×C10, C2×C4.Dic5, C2×D4⋊D5 [×2], D4.D10 [×8], C2×D4.D5 [×2], C2×C4○D20, D4×C2×C10, C2×D4.D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C8⋊C22, C2×C5⋊D4 [×6], C23×D5, D4.D10 [×2], C22×C5⋊D4, C2×D4.D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 4 4 20 20 2 2 2 2 20 20 2 2 20 20 20 20 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4.D10 kernel C2×D4.D10 C2×C4.Dic5 C2×D4⋊D5 D4.D10 C2×D4.D5 C2×C4○D20 D4×C2×C10 C2×C20 C22×C10 C22×D4 C22×C4 C2×D4 C2×C4 C23 C10 C2 # reps 1 1 2 8 2 1 1 3 1 2 2 12 12 4 2 8

Matrix representation of C2×D4.D10 in GL6(𝔽41)

 40 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 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 6 6 1 2 0 0 0 35 40 40
,
 40 0 0 0 0 0 11 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 23 0 0 0 0 18 0 0 0 0 0 14 14 16 32 0 0 21 6 25 25
,
 1 30 0 0 0 0 0 40 0 0 0 0 0 0 14 14 16 32 0 0 0 0 16 0 0 0 0 23 0 0 0 0 31 8 27 27

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,6,0,0,0,1,0,6,35,0,0,0,0,1,40,0,0,0,0,2,40],[40,11,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,14,21,0,0,23,0,14,6,0,0,0,0,16,25,0,0,0,0,32,25],[1,0,0,0,0,0,30,40,0,0,0,0,0,0,14,0,0,31,0,0,14,0,23,8,0,0,16,16,0,27,0,0,32,0,0,27] >;

C2×D4.D10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,1465);
// by ID

G=gap.SmallGroup(320,1465);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,1684,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=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,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^9>;
// generators/relations

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