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G = C2×D4⋊D5order 160 = 25·5

Direct product of C2 and D4⋊D5

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

Aliases: C2×D4⋊D5, C102D8, D43D10, C20.14D4, D205C22, C20.11C23, C53(C2×D8), (C2×D4)⋊1D5, (C2×D20)⋊8C2, (D4×C10)⋊1C2, C52C87C22, C10.44(C2×D4), (C2×C10).38D4, (C2×C4).47D10, (C5×D4)⋊3C22, C4.5(C5⋊D4), C4.11(C22×D5), (C2×C20).29C22, C22.21(C5⋊D4), (C2×C52C8)⋊4C2, C2.8(C2×C5⋊D4), SmallGroup(160,152)

Series: Derived Chief Lower central Upper central

C1C20 — C2×D4⋊D5
C1C5C10C20D20C2×D20 — C2×D4⋊D5
C5C10C20 — C2×D4⋊D5
C1C22C2×C4C2×D4

Generators and relations for C2×D4⋊D5
 G = < a,b,c,d,e | a2=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 280 in 76 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C5, C8 [×2], C2×C4, D4 [×2], D4 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8, D8 [×4], C2×D4, C2×D4, C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C2×D8, C52C8 [×2], D20 [×2], D20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×C10, C2×C52C8, D4⋊D5 [×4], C2×D20, D4×C10, C2×D4⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C5⋊D4 [×2], C22×D5, D4⋊D5 [×2], C2×C5⋊D4, C2×D4⋊D5

Smallest permutation representation of C2×D4⋊D5
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 76)(2 77)(3 78)(4 79)(5 80)(6 71)(7 72)(8 73)(9 74)(10 75)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 46)(32 47)(33 48)(34 49)(35 50)(36 41)(37 42)(38 43)(39 44)(40 45)
(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 75)(22 74)(23 73)(24 72)(25 71)(26 80)(27 79)(28 78)(29 77)(30 76)(31 65)(32 64)(33 63)(34 62)(35 61)(36 70)(37 69)(38 68)(39 67)(40 66)

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

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

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

C2×D4⋊D5 is a maximal subgroup of
D20.3D4  Dic54D8  D4⋊D20  D10⋊D8  D43D20  C5⋊(C82D4)  D4⋊D56C4  D203D4  D20.D4  C42.48D10  C207D8  D4.1D20  D2016D4  D2017D4  (C2×C10)⋊D8  C4⋊D4⋊D5  D20.23D4  C42.64D10  C42.214D10  C202D8  C20⋊D8  C42.74D10  Dic5⋊D8  C405D4  C4011D4  D20⋊D4  (C5×D4).D4  C40.43D4  D207D4  C409D4  M4(2).D10  (C2×C10)⋊8D8  (C5×D4)⋊14D4  C2×D5×D8  D85D10  D20.32C23
C2×D4⋊D5 is a maximal quotient of
(C2×C10).40D8  C20.50D8  C207D8  (C2×C10).D8  D2016D4  (C2×C10)⋊D8  C20.16D8  C202D8  C20⋊D8  C20.17D8  D206Q8  C20.D8  D8.D10  Q16.D10  D8⋊D10  C40.30C23  C40.31C23  (C2×C10)⋊8D8

34 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D
order122222224455888810···1010···1020202020
size11114420202222101010102···24···44444

34 irreducible representations

dim11111222222224
type++++++++++++
imageC1C2C2C2C2D4D4D5D8D10D10C5⋊D4C5⋊D4D4⋊D5
kernelC2×D4⋊D5C2×C52C8D4⋊D5C2×D20D4×C10C20C2×C10C2×D4C10C2×C4D4C4C22C2
# reps11411112424444

Matrix representation of C2×D4⋊D5 in GL4(𝔽41) generated by

40000
04000
00400
00040
,
1000
0100
00139
00140
,
40000
04000
00017
00290
,
354000
364000
0010
0001
,
0700
6000
00400
00401
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,40,0,0,0,0,0,29,0,0,17,0],[35,36,0,0,40,40,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,7,0,0,0,0,0,40,40,0,0,0,1] >;

C2×D4⋊D5 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_5
% in TeX

G:=Group("C2xD4:D5");
// GroupNames label

G:=SmallGroup(160,152);
// by ID

G=gap.SmallGroup(160,152);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,579,159,69,4613]);
// Polycyclic

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

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