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G = C207D4order 160 = 25·5

1st semidirect product of C20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C207D4, C221D20, C23.24D10, (C2×C10)⋊5D4, (C2×D20)⋊6C2, C53(C4⋊D4), C43(C5⋊D4), C4⋊Dic59C2, (C22×C4)⋊4D5, (C22×C20)⋊6C2, C2.17(C2×D20), C10.43(C2×D4), (C2×C4).85D10, D10⋊C43C2, C10.19(C4○D4), C2.19(C4○D20), (C2×C10).48C23, (C2×C20).94C22, C22.56(C22×D5), (C22×C10).40C22, (C2×Dic5).16C22, (C22×D5).10C22, (C2×C5⋊D4)⋊3C2, C2.7(C2×C5⋊D4), SmallGroup(160,151)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C207D4
C1C5C10C2×C10C22×D5C2×D20 — C207D4
C5C2×C10 — C207D4
C1C22C22×C4

Generators and relations for C207D4
 G = < a,b,c | a20=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 336 in 94 conjugacy classes, 37 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, C4⋊Dic5, D10⋊C4 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, C207D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, C2×D20, C4○D20, C2×C5⋊D4, C207D4

Smallest permutation representation of C207D4
On 80 points
Generators in S80
(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 27 50 76)(2 26 51 75)(3 25 52 74)(4 24 53 73)(5 23 54 72)(6 22 55 71)(7 21 56 70)(8 40 57 69)(9 39 58 68)(10 38 59 67)(11 37 60 66)(12 36 41 65)(13 35 42 64)(14 34 43 63)(15 33 44 62)(16 32 45 61)(17 31 46 80)(18 30 47 79)(19 29 48 78)(20 28 49 77)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)

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

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

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

C207D4 is a maximal subgroup of
(C2×D20)⋊C4  C22.2D40  D2013D4  D2014D4  C23.38D20  C22.D40  C23.13D20  Dic1014D4  (C2×C10).40D8  C4⋊C4.228D10  C4⋊C4.236D10  (C2×C10)⋊D8  C4⋊D4⋊D5  C52C824D4  C22⋊Q8⋊D5  C4030D4  C4029D4  C402D4  C403D4  (C2×C10)⋊8D8  (C5×Q8)⋊13D4  (C5×D4)⋊14D4  C42.276D10  C42.277D10  C24.27D10  C233D20  C24.30D10  C10.2- 1+4  C10.2+ 1+4  C10.112+ 1+4  C10.62- 1+4  C42.95D10  C42.97D10  C42.99D10  C42.100D10  C42.104D10  D4×D20  D2023D4  Dic1023D4  Dic1024D4  D45D20  D46D20  C4217D10  C42.116D10  C42.117D10  C42.119D10  C20⋊(C4○D4)  C10.682- 1+4  D5×C4⋊D4  C10.372+ 1+4  C10.382+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C22⋊Q825D5  C4⋊C426D10  C10.172- 1+4  C10.242- 1+4  C10.562+ 1+4  C10.572+ 1+4  C10.262- 1+4  C10.612+ 1+4  C10.662+ 1+4  C10.682+ 1+4  C10.692+ 1+4  C24.72D10  D4×C5⋊D4  C10.452- 1+4  C10.1452+ 1+4  C10.1462+ 1+4  C10.1472+ 1+4  C10.1482+ 1+4  D6⋊D20  C604D4  C606D4  (C2×C6)⋊D20  C6029D4  C20⋊S4
C207D4 is a maximal quotient of
C207(C4⋊C4)  (C2×C4)⋊6D20  (C2×C42)⋊D5  C23.14D20  C232D20  C24.16D10  (C2×C20).53D4  (C2×C4)⋊3D20  (C2×C20).56D4  C207D8  D4.1D20  D4.2D20  Q8⋊D20  Q8.1D20  C207Q16  C4030D4  C4029D4  C40.82D4  C402D4  C403D4  C40.4D4  D4.3D20  D4.4D20  D4.5D20  C24.64D10  C24.65D10  D6⋊D20  C604D4  C606D4  (C2×C6)⋊D20  C6029D4

46 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B10A···10N20A···20P
order122222224444445510···1020···20
size111122202022222020222···22···2

46 irreducible representations

dim111111222222222
type++++++++++++
imageC1C2C2C2C2C2D4D4D5C4○D4D10D10C5⋊D4D20C4○D20
kernelC207D4C4⋊Dic5D10⋊C4C2×D20C2×C5⋊D4C22×C20C20C2×C10C22×C4C10C2×C4C23C4C22C2
# reps112121222242888

Matrix representation of C207D4 in GL4(𝔽41) generated by

1100
333400
001430
00119
,
382000
20300
00400
0071
,
343500
8700
0010
003440
G:=sub<GL(4,GF(41))| [1,33,0,0,1,34,0,0,0,0,14,11,0,0,30,9],[38,20,0,0,20,3,0,0,0,0,40,7,0,0,0,1],[34,8,0,0,35,7,0,0,0,0,1,34,0,0,0,40] >;

C207D4 in GAP, Magma, Sage, TeX

C_{20}\rtimes_7D_4
% in TeX

G:=Group("C20:7D4");
// GroupNames label

G:=SmallGroup(160,151);
// by ID

G=gap.SmallGroup(160,151);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,4613]);
// Polycyclic

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

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