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G = C2×D10⋊C4order 160 = 25·5

Direct product of C2 and D10⋊C4

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

Aliases: C2×D10⋊C4, C23.31D10, C22.16D20, (C2×C4)⋊8D10, D107(C2×C4), C2.3(C2×D20), (C22×C4)⋊1D5, (C22×C20)⋊1C2, C10.40(C2×D4), (C2×C10).36D4, (C22×D5)⋊4C4, C102(C22⋊C4), (C2×C20)⋊10C22, (C23×D5).2C2, C22.17(C4×D5), (C2×C10).45C23, C10.31(C22×C4), (C2×Dic5)⋊6C22, (C22×Dic5)⋊3C2, C22.20(C5⋊D4), C22.23(C22×D5), (C22×C10).37C22, (C22×D5).26C22, C53(C2×C22⋊C4), C2.19(C2×C4×D5), C2.2(C2×C5⋊D4), (C2×C10).38(C2×C4), SmallGroup(160,148)

Series: Derived Chief Lower central Upper central

C1C10 — C2×D10⋊C4
C1C5C10C2×C10C22×D5C23×D5 — C2×D10⋊C4
C5C10 — C2×D10⋊C4
C1C23C22×C4

Generators and relations for C2×D10⋊C4
 G = < a,b,c,d | a2=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >

Subgroups: 456 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C22, C22 [×6], C22 [×16], C5, C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×4], C22×C4, C22×C4, C24, Dic5 [×2], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×6], C22×D5 [×4], C22×C10, D10⋊C4 [×4], C22×Dic5, C22×C20, C23×D5, C2×D10⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4

Smallest permutation representation of C2×D10⋊C4
On 80 points
Generators in S80
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 76)(22 77)(23 78)(24 79)(25 80)(26 71)(27 72)(28 73)(29 74)(30 75)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 60)(8 59)(9 58)(10 57)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 80)(29 79)(30 78)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 70)(39 69)(40 68)
(1 37 17 27)(2 38 18 28)(3 39 19 29)(4 40 20 30)(5 31 11 21)(6 32 12 22)(7 33 13 23)(8 34 14 24)(9 35 15 25)(10 36 16 26)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)

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

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

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

C2×D10⋊C4 is a maximal subgroup of
C53(C23⋊C8)  C5⋊(C23⋊C8)  C22⋊F5⋊C4  C22.58(D4×D5)  (C2×C4)⋊9D20  D102C42  D102(C4⋊C4)  D103(C4⋊C4)  C10.54(C4×D4)  C10.55(C4×D4)  (C2×C20)⋊5D4  (C2×Dic5)⋊3D4  (C2×C4).20D20  (C2×C4).21D20  C10.(C4⋊D4)  (C22×D5).Q8  (C2×C20).33D4  (C2×C4)⋊6D20  (C2×C42)⋊D5  C24.48D10  C24.12D10  C24.13D10  C23.45D20  C24.14D10  C232D20  C24.16D10  D104(C4⋊C4)  (C2×D20)⋊22C4  D105(C4⋊C4)  C10.90(C4×D4)  (C2×C4)⋊3D20  (C2×C20).289D4  (C2×C20).290D4  (C2×C20).56D4  C24.65D10  C24.21D10  (C22×D5)⋊Q8  C2×C4×D20  C2×D5×C22⋊C4  C4210D10  C4211D10  D45D20  C4216D10  C4217D10  C10.372+ 1+4  C10.402+ 1+4  C10.462+ 1+4  C10.512+ 1+4  C10.532+ 1+4  C10.562+ 1+4  C10.1212+ 1+4  C10.1222+ 1+4  C2×C4×C5⋊D4  C10.1452+ 1+4
C2×D10⋊C4 is a maximal quotient of
(C2×C20)⋊10Q8  (C2×C4)⋊6D20  C23.42D20  C24.48D10  C23.45D20  C4○D209C4  (C2×Dic5)⋊6Q8  D104(C4⋊C4)  (C2×D20)⋊22C4  C4⋊C436D10  C4○D2010C4  C4.(C2×D20)  C424D10  (C2×D20)⋊25C4  (C22×C8)⋊D5  C23.23D20  C23.46D20  D108M4(2)  C4.89(C2×D20)  C23.48D20  M4(2).31D10  C23.49D20  C23.20D20  C24.65D10

52 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H5A5B10A···10N20A···20P
order12···22222444444445510···1020···20
size11···110101010222210101010222···22···2

52 irreducible representations

dim1111112222222
type++++++++++
imageC1C2C2C2C2C4D4D5D10D10C4×D5D20C5⋊D4
kernelC2×D10⋊C4D10⋊C4C22×Dic5C22×C20C23×D5C22×D5C2×C10C22×C4C2×C4C23C22C22C22
# reps1411184242888

Matrix representation of C2×D10⋊C4 in GL5(𝔽41)

400000
040000
004000
00010
00001
,
10000
063400
06000
000400
000040
,
10000
040100
00100
00010
000040
,
320000
040000
004000
00001
00010

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,6,6,0,0,0,34,0,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,40],[32,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×D10⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes C_4
% in TeX

G:=Group("C2xD10:C4");
// GroupNames label

G:=SmallGroup(160,148);
// by ID

G=gap.SmallGroup(160,148);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations

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