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G = (C2xD20):C4order 320 = 26·5

1st semidirect product of C2xD20 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2xD20):1C4, C4:Dic5:1C4, C10.13C4wrC2, (C2xC10).29D8, (C2xC4).103D20, C20:7D4.7C2, (C2xC20).221D4, C20.55D4:1C2, C22.7(D4:D5), (C2xC10).37SD16, C2.C42:6D5, (C22xC4).52D10, C5:3(C22.SD16), C22.7(Q8:D5), C2.3(D20:6C4), C2.4(D20:4C4), C10.22(C23:C4), (C22xC10).174D4, C23.70(C5:D4), C10.16(D4:C4), (C22xC20).89C22, C2.4(C23.1D10), C22.54(D10:C4), (C2xC4).8(C4xD5), (C2xC20).193(C2xC4), (C2xC10).97(C22:C4), (C5xC2.C42):13C2, SmallGroup(320,9)

Series: Derived Chief Lower central Upper central

C1C2xC20 — (C2xD20):C4
C1C5C10C2xC10C22xC10C22xC20C20:7D4 — (C2xD20):C4
C5C2xC10C2xC20 — (C2xD20):C4
C1C22C22xC4C2.C42

Generators and relations for (C2xD20):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, C2, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, C23, C23, D5, C10, C10, C22:C4, C4:C4, C2xC8, C22xC4, C22xC4, C2xD4, Dic5, C20, D10, C2xC10, C2xC10, C2.C42, C22:C8, C4:D4, C5:2C8, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C22xD5, C22xC10, C22.SD16, C2xC5:2C8, C4:Dic5, D10:C4, C2xD20, C2xC5:D4, C22xC20, C22xC20, C20.55D4, C5xC2.C42, C20:7D4, (C2xD20):C4
Quotients: C1, C2, C4, C22, C2xC4, D4, D5, C22:C4, D8, SD16, D10, C23:C4, D4:C4, C4wrC2, C4xD5, D20, C5:D4, C22.SD16, D10:C4, D4:D5, Q8:D5, D20:4C4, C23.1D10, D20:6C4, (C2xD20):C4

Smallest permutation representation of (C2xD20):C4
On 80 points
Generators in S80
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 60)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)
(2 70)(4 72)(6 74)(8 76)(10 78)(12 80)(14 62)(16 64)(18 66)(20 68)(21 45 60 26)(22 37 41 56)(23 47 42 28)(24 39 43 58)(25 49 44 30)(27 51 46 32)(29 53 48 34)(31 55 50 36)(33 57 52 38)(35 59 54 40)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C···4G4H5A5B8A8B8C8D10A···10N20A···20X
order1222222444···4455888810···1020···20
size11112240224···44022202020202···24···4

59 irreducible representations

dim111111222222222224444
type+++++++++++++
imageC1C2C2C2C4C4D4D4D5D8SD16D10C4wrC2C4xD5D20C5:D4D20:4C4C23:C4D4:D5Q8:D5C23.1D10
kernel(C2xD20):C4C20.55D4C5xC2.C42C20:7D4C4:Dic5C2xD20C2xC20C22xC10C2.C42C2xC10C2xC10C22xC4C10C2xC4C2xC4C23C2C10C22C22C2
# reps1111221122224444161224

Matrix representation of (C2xD20):C4 in GL6(F41)

100000
010000
0040000
0004000
000010
000001
,
32260000
090000
009000
00383200
0000100
00002937
,
14160000
16270000
00382300
005300
0000540
00002436
,
100000
7400000
001000
0015900
000010
000001

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] >;

(C2xD20):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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