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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), (D4×D5)⋊5C22, D4⋊2(C22×D5), (C5×D4)⋊2C23, 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×C10)⋊19C22, D4⋊2D5⋊5C22, 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
Upper central
C1 — C22 — C2×C4 — C2×D8

Generators and relations for C2×D8⋊D5
 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 >

Subgroups: 1310 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), D8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×M4(2), C2×D8, C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C2×C8⋊C22, C8⋊D5, C40⋊C2, C2×C5⋊2C8, D4⋊D5, D4.D5, C2×C40, C5×D8, C2×Dic10, C2×C4×D5, C2×D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C23×D5, C2×C8⋊D5, C2×C40⋊C2, D8⋊D5, C2×D4⋊D5, C2×D4.D5, C10×D8, C2×D4×D5, C2×D4⋊2D5, C2×D8⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8⋊C22, C22×D4, C22×D5, C2×C8⋊C22, D4×D5, C23×D5, D8⋊D5, C2×D4×D5, C2×D8⋊D5

Smallest permutation representation of C2×D8⋊D5
►On 80 points
Generators in S80
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 65)(32 66)(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 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 31)(26 30)(27 29)(33 39)(34 38)(35 37)(42 48)(43 47)(44 46)(49 51)(52 56)(53 55)(58 64)(59 63)(60 62)(65 67)(68 72)(69 71)(74 80)(75 79)(76 78)

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

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

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

Matrix representation of C2×D8⋊D5 ►in 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] >;
 

C2×D8⋊D5 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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