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G = C2xD5:C8order 160 = 25·5

Direct product of C2 and D5:C8

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2xD5:C8, D10:3C8, Dic5.9C23, D5:(C2xC8), C10:1(C2xC8), C5:C8:4C22, C5:1(C22xC8), (C4xD5).7C4, C4.20(C2xF5), (C2xC4).11F5, (C2xC20).11C4, C20.19(C2xC4), C2.1(C22xF5), D10.12(C2xC4), C10.1(C22xC4), (C22xD5).7C4, C22.15(C2xF5), Dic5.14(C2xC4), (C4xD5).33C22, (C2xDic5).54C22, (C2xC5:C8):6C2, (C2xC4xD5).17C2, (C2xC10).13(C2xC4), SmallGroup(160,200)

Series: Derived Chief Lower central Upper central

C1C5 — C2xD5:C8
C1C5C10Dic5C5:C8C2xC5:C8 — C2xD5:C8
C5 — C2xD5:C8
C1C2xC4

Generators and relations for C2xD5:C8
 G = < a,b,c,d | a2=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Subgroups: 196 in 76 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, C23, D5, C10, C10, C2xC8, C22xC4, Dic5, C20, D10, C2xC10, C22xC8, C5:C8, C4xD5, C2xDic5, C2xC20, C22xD5, D5:C8, C2xC5:C8, C2xC4xD5, C2xD5:C8
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C2xC8, C22xC4, F5, C22xC8, C2xF5, D5:C8, C22xF5, C2xD5:C8

Smallest permutation representation of C2xD5:C8
On 80 points
Generators in S80
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 77)(26 78)(27 79)(28 80)(29 73)(30 74)(31 75)(32 76)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(1 21 75 33 57)(2 34 22 58 76)(3 59 35 77 23)(4 78 60 24 36)(5 17 79 37 61)(6 38 18 62 80)(7 63 39 73 19)(8 74 64 20 40)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(1 55)(2 32)(3 13)(4 42)(5 51)(6 28)(7 9)(8 46)(10 74)(11 33)(12 22)(14 78)(15 37)(16 18)(17 43)(19 71)(20 30)(21 47)(23 67)(24 26)(25 59)(27 79)(29 63)(31 75)(34 56)(35 41)(36 68)(38 52)(39 45)(40 72)(44 62)(48 58)(49 77)(50 60)(53 73)(54 64)(57 65)(61 69)(66 76)(70 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)

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

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

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

C2xD5:C8 is a maximal subgroup of
D10.18D8  D10.3M4(2)  M4(2).F5  C42.5F5  C42.11F5  C5:C8:8D4  D10:M4(2)  D10.C42  D20:2C8  D10:2M4(2)  C4:C4.9F5  C2xC8xF5  M4(2):5F5  D10.11M4(2)  (C2xD4).8F5  (C2xQ8).5F5  C4oD20:C4  Dic5.21C24
C2xD5:C8 is a maximal quotient of
C42.6F5  C42.11F5  C42.12F5  C5:C8:8D4  D20:2C8  Dic10:C8  D5:M5(2)  Dic10.C8  C2xC4xC5:C8  D10.11M4(2)  C20.34M4(2)

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H 5 8A···8P10A10B10C20A20B20C20D
order122222224444444458···810101020202020
size111155551111555545···54444444

40 irreducible representations

dim111111114444
type+++++++
imageC1C2C2C2C4C4C4C8F5C2xF5C2xF5D5:C8
kernelC2xD5:C8D5:C8C2xC5:C8C2xC4xD5C4xD5C2xC20C22xD5D10C2xC4C4C22C2
# reps1421422161214

Matrix representation of C2xD5:C8 in GL5(F41)

400000
040000
004000
000400
000040
,
10000
000040
010040
001040
000140
,
10000
000401
004001
040001
00001
,
400000
01130120
02330011
01103023
00123011

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[1,0,0,0,0,0,0,0,40,0,0,0,40,0,0,0,40,0,0,0,0,1,1,1,1],[40,0,0,0,0,0,11,23,11,0,0,30,30,0,12,0,12,0,30,30,0,0,11,23,11] >;

C2xD5:C8 in GAP, Magma, Sage, TeX

C_2\times D_5\rtimes C_8
% in TeX

G:=Group("C2xD5:C8");
// GroupNames label

G:=SmallGroup(160,200);
// by ID

G=gap.SmallGroup(160,200);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,69,2309,599]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations

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