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G = C2×Q8×D5order 160 = 25·5

Direct product of C2, Q8 and D5

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

Aliases: C2×Q8×D5, C10.8C24, C20.22C23, Dic109C22, D10.17C23, Dic5.5C23, C102(C2×Q8), C52(C22×Q8), (Q8×C10)⋊5C2, (C2×C4).61D10, (C5×Q8)⋊5C22, C2.9(C23×D5), C4.22(C22×D5), (C2×Dic10)⋊13C2, (C2×C20).46C22, (C2×C10).66C23, (C4×D5).21C22, C22.31(C22×D5), (C2×Dic5).47C22, (C22×D5).45C22, (C2×C4×D5).6C2, SmallGroup(160,220)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×Q8×D5
 G = < a,b,c,d,e | a2=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 392 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C4 [×6], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×15], Q8 [×4], Q8 [×12], C23, D5 [×4], C10, C10 [×2], C22×C4 [×3], C2×Q8, C2×Q8 [×11], Dic5 [×6], C20 [×6], D10 [×6], C2×C10, C22×Q8, Dic10 [×12], C4×D5 [×12], C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C22×D5, C2×Dic10 [×3], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, C2×Q8×D5
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, C22×D5 [×7], Q8×D5 [×2], C23×D5, C2×Q8×D5

Smallest permutation representation of C2×Q8×D5
On 80 points
Generators in S80
(1 44)(2 45)(3 41)(4 42)(5 43)(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 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(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 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(11 41 16 46)(12 42 17 47)(13 43 18 48)(14 44 19 49)(15 45 20 50)(21 71 26 76)(22 72 27 77)(23 73 28 78)(24 74 29 79)(25 75 30 80)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(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 43)(2 42)(3 41)(4 45)(5 44)(6 46)(7 50)(8 49)(9 48)(10 47)(11 51)(12 55)(13 54)(14 53)(15 52)(16 56)(17 60)(18 59)(19 58)(20 57)(21 61)(22 65)(23 64)(24 63)(25 62)(26 66)(27 70)(28 69)(29 68)(30 67)(31 71)(32 75)(33 74)(34 73)(35 72)(36 76)(37 80)(38 79)(39 78)(40 77)

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

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

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

C2×Q8×D5 is a maximal subgroup of
(Q8×D5)⋊C4  Q82D20  D104Q16  D108SD16  D105Q16  (C2×Q8)⋊4F5  (C2×Q8).7F5  (C2×F5)⋊Q8  C42.125D10  Q85D20  C10.162- 1+4  Dic1021D4  Dic1022D4  C42.141D10  Dic1010D4  C42.171D10  D208Q8  C10.1072- 1+4  D5.2- 1+4
C2×Q8×D5 is a maximal quotient of
C10.102+ 1+4  Dic1010Q8  C42.232D10  D2010Q8  (Q8×Dic5)⋊C2  C10.502+ 1+4  Dic1021D4  C10.512+ 1+4  C10.1182+ 1+4  C10.522+ 1+4  Dic107Q8  C42.236D10  C42.148D10  D207Q8  Dic108Q8  Dic109Q8  D208Q8  C42.241D10  C42.174D10  D209Q8

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G···4L5A5B10A···10F20A···20L
order122222224···44···45510···1020···20
size111155552···210···10222···24···4

40 irreducible representations

dim1111122224
type+++++-+++-
imageC1C2C2C2C2Q8D5D10D10Q8×D5
kernelC2×Q8×D5C2×Dic10C2×C4×D5Q8×D5Q8×C10D10C2×Q8C2×C4Q8C2
# reps1338142684

Matrix representation of C2×Q8×D5 in GL4(𝔽41) generated by

40000
04000
00400
00040
,
40000
04000
00923
00032
,
40000
04000
0090
00932
,
0100
40600
0010
0001
,
04000
40000
00400
00040
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,23,32],[40,0,0,0,0,40,0,0,0,0,9,9,0,0,0,32],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,40,0,0,0,0,40] >;

C2×Q8×D5 in GAP, Magma, Sage, TeX

C_2\times Q_8\times D_5
% in TeX

G:=Group("C2xQ8xD5");
// GroupNames label

G:=SmallGroup(160,220);
// by ID

G=gap.SmallGroup(160,220);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,159,69,4613]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
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