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G = C2×Q82D5order 160 = 25·5

Direct product of C2 and Q82D5

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

Aliases: C2×Q82D5, Q85D10, D209C22, C10.9C24, C20.23C23, D10.4C23, Dic5.16C23, (C2×Q8)⋊6D5, (Q8×C10)⋊6C2, (C2×D20)⋊12C2, C103(C4○D4), (C2×C4).62D10, (C4×D5)⋊5C22, (C5×Q8)⋊6C22, C4.23(C22×D5), C2.10(C23×D5), (C2×C20).47C22, (C2×C10).67C23, C22.32(C22×D5), (C2×Dic5).65C22, (C22×D5).33C22, (C2×C4×D5)⋊5C2, C53(C2×C4○D4), SmallGroup(160,221)

Series: Derived Chief Lower central Upper central

C1C10 — C2×Q82D5
C1C5C10D10C22×D5C2×C4×D5 — C2×Q82D5
C5C10 — C2×Q82D5
C1C22C2×Q8

Generators and relations for C2×Q82D5
 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 [×2], C2 [×6], C4 [×6], C4 [×2], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10, C10 [×2], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×6], D10 [×6], D10 [×6], C2×C10, C2×C4○D4, C4×D5 [×12], D20 [×12], C2×Dic5, C2×C20 [×3], C5×Q8 [×4], C22×D5 [×3], C2×C4×D5 [×3], C2×D20 [×3], Q82D5 [×8], Q8×C10, C2×Q82D5
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, C22×D5 [×7], Q82D5 [×2], C23×D5, C2×Q82D5

Smallest permutation representation of C2×Q82D5
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)])

C2×Q82D5 is a maximal subgroup of
M4(2).21D10  Q8⋊(C4×D5)  Q82D5⋊C4  Q8.D20  D204D4  D207D4  D20.17D4  (C2×Q8)⋊6F5  (C2×Q8)⋊7F5  (C2×Q8).5F5  C42.126D10  Q85D20  Q86D20  C4⋊C426D10  C10.172- 1+4  D2021D4  D2022D4  C42.233D10  C4218D10  D2010D4  C42.171D10  C42.240D10  D2012D4  D40⋊C22  C10.452- 1+4  C10.1482+ 1+4  Dic5.20C24  C2×D5×C4○D4  D20.39C23
C2×Q82D5 is a maximal quotient of
C10.112+ 1+4  Q86Dic10  Q86D20  C42.131D10  C42.135D10  C42.136D10  C22⋊Q825D5  C4⋊C426D10  D2021D4  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  D2012D4  C42.241D10  D209Q8  C42.177D10  C42.178D10  C42.179D10  C2×Q8×Dic5  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+++++++++
imageC1C2C2C2C2D5C4○D4D10D10Q82D5
kernelC2×Q82D5C2×C4×D5C2×D20Q82D5Q8×C10C2×Q8C10C2×C4Q8C2
# reps1338124684

Matrix representation of C2×Q82D5 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] >;

C2×Q82D5 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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