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## G = (C2×D20)⋊C4order 320 = 26·5

### 1st semidirect product of C2×D20 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×D20)⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×C20 — C20⋊7D4 — (C2×D20)⋊C4
 Lower central C5 — C2×C10 — C2×C20 — (C2×D20)⋊C4
 Upper central C1 — C22 — C22×C4 — C2.C42

Generators and relations for (C2×D20)⋊C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=ab11, dcd-1=b15c >

Subgroups: 430 in 90 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22 [×3], C22 [×5], C5, C8, C2×C4 [×2], C2×C4 [×6], D4 [×3], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4 [×2], Dic5, C20 [×4], D10 [×3], C2×C10 [×3], C2×C10 [×2], C2.C42, C22⋊C8, C4⋊D4, C52C8, D20, C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×5], C22×D5, C22×C10, C22.SD16, C2×C52C8, C4⋊Dic5, D10⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, C20.55D4, C5×C2.C42, C207D4, (C2×D20)⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16, D10, C23⋊C4, D4⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, C22.SD16, D10⋊C4, D4⋊D5, Q8⋊D5, D204C4, C23.1D10, D206C4, (C2×D20)⋊C4

Smallest permutation representation of (C2×D20)⋊C4
On 80 points
Generators in S80
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 61)(20 62)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 41)
(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 44)(2 43)(3 42)(4 41)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 65)(22 64)(23 63)(24 62)(25 61)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 72)(35 71)(36 70)(37 69)(38 68)(39 67)(40 66)
(2 74)(4 76)(6 78)(8 80)(10 62)(12 64)(14 66)(16 68)(18 70)(20 72)(21 47 42 26)(22 37 43 58)(23 49 44 28)(24 39 45 60)(25 51 46 30)(27 53 48 32)(29 55 50 34)(31 57 52 36)(33 59 54 38)(35 41 56 40)

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 40 2 2 4 ··· 4 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D5 D8 SD16 D10 C4≀C2 C4×D5 D20 C5⋊D4 D20⋊4C4 C23⋊C4 D4⋊D5 Q8⋊D5 C23.1D10 kernel (C2×D20)⋊C4 C20.55D4 C5×C2.C42 C20⋊7D4 C4⋊Dic5 C2×D20 C2×C20 C22×C10 C2.C42 C2×C10 C2×C10 C22×C4 C10 C2×C4 C2×C4 C23 C2 C10 C22 C22 C2 # reps 1 1 1 1 2 2 1 1 2 2 2 2 4 4 4 4 16 1 2 2 4

Matrix representation of (C2×D20)⋊C4 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
,
 32 26 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 38 32 0 0 0 0 0 0 10 0 0 0 0 0 29 37
,
 14 16 0 0 0 0 16 27 0 0 0 0 0 0 38 23 0 0 0 0 5 3 0 0 0 0 0 0 5 40 0 0 0 0 24 36
,
 1 0 0 0 0 0 7 40 0 0 0 0 0 0 1 0 0 0 0 0 15 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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],[32,0,0,0,0,0,26,9,0,0,0,0,0,0,9,38,0,0,0,0,0,32,0,0,0,0,0,0,10,29,0,0,0,0,0,37],[14,16,0,0,0,0,16,27,0,0,0,0,0,0,38,5,0,0,0,0,23,3,0,0,0,0,0,0,5,24,0,0,0,0,40,36],[1,7,0,0,0,0,0,40,0,0,0,0,0,0,1,15,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

(C2×D20)⋊C4 in GAP, Magma, Sage, TeX

(C_2\times D_{20})\rtimes C_4
% in TeX

G:=Group("(C2xD20):C4");
// GroupNames label

G:=SmallGroup(320,9);
// by ID

G=gap.SmallGroup(320,9);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,1571,570,192,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a*b^11,d*c*d^-1=b^15*c>;
// generators/relations

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