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G = C2xQ8:2D5order 160 = 25·5

Direct product of C2 and Q8:2D5

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

Aliases: C2xQ8:2D5, Q8:5D10, D20:9C22, C10.9C24, C20.23C23, D10.4C23, Dic5.16C23, (C2xQ8):6D5, (Q8xC10):6C2, (C2xD20):12C2, C10:3(C4oD4), (C2xC4).62D10, (C4xD5):5C22, (C5xQ8):6C22, C4.23(C22xD5), C2.10(C23xD5), (C2xC20).47C22, (C2xC10).67C23, C22.32(C22xD5), (C2xDic5).65C22, (C22xD5).33C22, (C2xC4xD5):5C2, C5:3(C2xC4oD4), SmallGroup(160,221)

Series: Derived Chief Lower central Upper central

C1C10 — C2xQ8:2D5
C1C5C10D10C22xD5C2xC4xD5 — C2xQ8:2D5
C5C10 — C2xQ8:2D5
C1C22C2xQ8

Generators and relations for C2xQ8:2D5
 G = < a,b,c,d,e | a2=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 488 in 164 conjugacy classes, 89 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, C22xC4, C2xD4, C2xQ8, C4oD4, Dic5, C20, D10, D10, C2xC10, C2xC4oD4, C4xD5, D20, C2xDic5, C2xC20, C5xQ8, C22xD5, C2xC4xD5, C2xD20, Q8:2D5, Q8xC10, C2xQ8:2D5
Quotients: C1, C2, C22, C23, D5, C4oD4, C24, D10, C2xC4oD4, C22xD5, Q8:2D5, C23xD5, C2xQ8:2D5

Smallest permutation representation of C2xQ8:2D5
On 80 points
Generators in S80
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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 45)(2 44)(3 43)(4 42)(5 41)(6 50)(7 49)(8 48)(9 47)(10 46)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 65)(22 64)(23 63)(24 62)(25 61)(26 70)(27 69)(28 68)(29 67)(30 66)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)

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

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

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

C2xQ8:2D5 is a maximal subgroup of
M4(2).21D10  Q8:(C4xD5)  Q8:2D5:C4  Q8.D20  D20:4D4  D20:7D4  D20.17D4  (C2xQ8):6F5  (C2xQ8):7F5  (C2xQ8).5F5  C42.126D10  Q8:5D20  Q8:6D20  C4:C4:26D10  C10.172- 1+4  D20:21D4  D20:22D4  C42.233D10  C42:18D10  D20:10D4  C42.171D10  C42.240D10  D20:12D4  D40:C22  C10.452- 1+4  C10.1482+ 1+4  Dic5.20C24  C2xD5xC4oD4  D20.39C23
C2xQ8:2D5 is a maximal quotient of
C10.112+ 1+4  Q8:6Dic10  Q8:6D20  C42.131D10  C42.135D10  C42.136D10  C22:Q8:25D5  C4:C4:26D10  D20:21D4  C10.532+ 1+4  C10.772- 1+4  C10.562+ 1+4  C10.572+ 1+4  C42.237D10  C42.152D10  C42.153D10  C42.155D10  C42.156D10  C42.240D10  D20:12D4  C42.241D10  D20:9Q8  C42.177D10  C42.178D10  C42.179D10  C2xQ8xDic5  C10.452- 1+4

40 conjugacy classes

class 1 2A2B2C2D···2I4A···4F4G4H4I4J5A5B10A···10F20A···20L
order12222···24···444445510···1020···20
size111110···102···25555222···24···4

40 irreducible representations

dim1111122224
type+++++++++
imageC1C2C2C2C2D5C4oD4D10D10Q8:2D5
kernelC2xQ8:2D5C2xC4xD5C2xD20Q8:2D5Q8xC10C2xQ8C10C2xC4Q8C2
# reps1338124684

Matrix representation of C2xQ8:2D5 in GL4(F41) generated by

40000
04000
0010
0001
,
1000
0100
004039
0011
,
1000
0100
00320
0099
,
64000
1000
0010
0001
,
35100
6600
0010
004040
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,1,0,0,39,1],[1,0,0,0,0,1,0,0,0,0,32,9,0,0,0,9],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[35,6,0,0,1,6,0,0,0,0,1,40,0,0,0,40] >;

C2xQ8:2D5 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_2D_5
% in TeX

G:=Group("C2xQ8:2D5");
// GroupNames label

G:=SmallGroup(160,221);
// by ID

G=gap.SmallGroup(160,221);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,579,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^5=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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