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G = C2×D8⋊D5  order 320 = 26·5

Direct product of C2 and D8⋊D5

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

Aliases: C2×D8⋊D5, D8⋊9D10, C40⋊5C23, C20.2C24, D20.1C23, Dic10⋊1C23, (C2×C8)⋊8D10, (C2×D8)⋊11D5, C4.40(D4×D5), C8⋊3(C22×D5), (C10×D8)⋊11C2, (C2×D4)⋊28D10, C5⋊2C8⋊1C23, D4⋊D5⋊9C22, (C4×D5).14D4, C20.77(C2×D4), (C5×D4)⋊2C23, (D4×D5)⋊5C22, D4⋊2(C22×D5), C4.2(C23×D5), C10⋊2(C8⋊C22), (C2×C40)⋊16C22, D10.83(C2×D4), (C5×D8)⋊14C22, D4.D5⋊7C22, (C4×D5).1C23, D4⋊2D5⋊5C22, (D4×C10)⋊19C22, C40⋊C2⋊13C22, C8⋊D5⋊12C22, Dic5.94(C2×D4), C22.136(D4×D5), (C2×C20).519C23, (C2×Dic5).247D4, (C22×D5).134D4, C10.103(C22×D4), (C2×Dic10)⋊36C22, (C2×D20).182C22, (C2×D4×D5)⋊22C2, C5⋊2(C2×C8⋊C22), C2.76(C2×D4×D5), (C2×D4⋊D5)⋊26C2, (C2×C8⋊D5)⋊8C2, (C2×C40⋊C2)⋊24C2, (C2×D4.D5)⋊25C2, (C2×D4⋊2D5)⋊23C2, (C2×C5⋊2C8)⋊14C22, (C2×C10).392(C2×D4), (C2×C4×D5).164C22, (C2×C4).609(C22×D5), SmallGroup(320,1427)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — C2×D8⋊D5
Chief series
C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4×D5 — C2×D8⋊D5
Lower central
C5 — C10 — C20 — C2×D8⋊D5

Subgroups: 1310 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×24], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×4], D4 [×13], Q8 [×3], C23 [×12], D5 [×4], C10, C10 [×2], C10 [×4], C2×C8, C2×C8, M4(2) [×4], D8 [×4], D8 [×4], SD16 [×8], C22×C4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×6], C24, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×8], C2×M4(2), C2×D8, C2×D8, C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C5⋊2C8 [×2], C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×D5, C22×D5 [×9], C22×C10 [×2], C2×C8⋊C22, C8⋊D5 [×4], C40⋊C2 [×4], C2×C5⋊2C8, D4⋊D5 [×4], D4.D5 [×4], C2×C40, C5×D8 [×4], C2×Dic10, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], D4⋊2D5 [×4], D4⋊2D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], D4×C10 [×2], C23×D5, C2×C8⋊D5, C2×C40⋊C2, D8⋊D5 [×8], C2×D4⋊D5, C2×D4.D5, C10×D8, C2×D4×D5, C2×D4⋊2D5, C2×D8⋊D5

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C22×D5 [×7], C2×C8⋊C22, D4×D5 [×2], C23×D5, D8⋊D5 [×2], C2×D4×D5, C2×D8⋊D5

Generators and relations
 G = < a,b,c,d,e | a2=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

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

(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 7)(2 6)(3 5)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(58 64)(59 63)(60 62)(65 69)(66 68)(70 72)(73 75)(76 80)(77 79)

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
400200000
040020000
400100000
040010000
0000001031
000000039
00004211920
00000212122
,
400000000
040000000
400100000
040010000
00004002121
00000404038
00000010
00000001
,
640000000
10000000
006400000
00100000
0000404000
00008700
000000040
00000016
,
351000000
66000000
003510000
00660000
000003500
000034000
00000061
000000635

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,2,0,1,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,21,21,0,0,0,0,10,0,19,21,0,0,0,0,31,39,20,22],[40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,21,40,1,0,0,0,0,0,21,38,0,1],[6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,8,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6],[35,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,35,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35] >;
 

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D40A···40H
order12222222222244444455888810···1010···102020202040···40
size11114444101020202210102020224420202···28···844444···4

50 irreducible representations

dim11111111122222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10C8⋊C22D4×D5D4×D5D8⋊D5
kernelC2×D8⋊D5C2×C8⋊D5C2×C40⋊C2D8⋊D5C2×D4⋊D5C2×D4.D5C10×D8C2×D4×D5C2×D4⋊2D5C4×D5C2×Dic5C22×D5C2×D8C2×C8D8C2×D4C10C4C22C2
# reps11181111121122842228

In GAP, Magma, Sage, TeX 

C_2\times D_8\rtimes D_5
 
% in TeX
 
G:=Group("C2xD8:D5");
 
// GroupNames label
 
G:=SmallGroup(320,1427);
 
// by ID
 
G=gap.SmallGroup(320,1427);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,185,438,235,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^8=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=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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