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## G = C2×C20.46D4order 320 = 26·5

### Direct product of C2 and C20.46D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20.46D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C22×D20 — C2×C20.46D4
 Lower central C5 — C10 — C2×C10 — C2×C20.46D4
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C20.46D4
G = < a,b,c,d | a2=b20=d2=1, c4=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b15c3 >

Subgroups: 958 in 186 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C24, C20, D10, C2×C10, C2×C10, C4.D4, C2×M4(2), C2×M4(2), C22×D4, C52C8, C40, D20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C4.D4, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C2×D20, C2×D20, C22×C20, C23×D5, C20.46D4, C2×C4.Dic5, C10×M4(2), C22×D20, C2×C20.46D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4.D4, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C2×C4.D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C20.46D4, C2×D10⋊C4, C2×C20.46D4

Smallest permutation representation of C2×C20.46D4
On 80 points
Generators in S80
(1 58)(2 59)(3 60)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 61)
(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 31 53 77 11 21 43 67)(2 30 54 76 12 40 44 66)(3 29 55 75 13 39 45 65)(4 28 56 74 14 38 46 64)(5 27 57 73 15 37 47 63)(6 26 58 72 16 36 48 62)(7 25 59 71 17 35 49 61)(8 24 60 70 18 34 50 80)(9 23 41 69 19 33 51 79)(10 22 42 68 20 32 52 78)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 20 20 20 20 2 2 2 2 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 C4×D5 C4.D4 C20.46D4 kernel C2×C20.46D4 C20.46D4 C2×C4.Dic5 C10×M4(2) C22×D20 C2×D20 C23×D5 C2×C20 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C10 C2 # reps 1 4 1 1 1 4 4 4 2 4 2 4 8 8 4 2 8

Matrix representation of C2×C20.46D4 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
,
 6 1 0 0 0 0 40 0 0 0 0 0 0 0 9 11 0 0 0 0 30 14 0 0 0 0 0 0 39 11 0 0 0 0 14 25
,
 6 35 0 0 0 0 40 35 0 0 0 0 0 0 2 5 4 9 0 0 32 39 13 28 0 0 4 21 12 8 0 0 38 17 23 29
,
 6 35 0 0 0 0 40 35 0 0 0 0 0 0 11 9 0 0 0 0 14 30 0 0 0 0 29 39 25 30 0 0 29 10 12 16

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],[6,40,0,0,0,0,1,0,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,2,32,4,38,0,0,5,39,21,17,0,0,4,13,12,23,0,0,9,28,8,29],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,11,14,29,29,0,0,9,30,39,10,0,0,0,0,25,12,0,0,0,0,30,16] >;

C2×C20.46D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{46}D_4
% in TeX

G:=Group("C2xC20.46D4");
// GroupNames label

G:=SmallGroup(320,757);
// by ID

G=gap.SmallGroup(320,757);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,136,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=d^2=1,c^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^15*c^3>;
// generators/relations

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