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G = D4013C4order 320 = 26·5

7th semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4013C4, Dic2013C4, M4(2).27D10, C56(C8○D8), C8.16(C4×D5), C40⋊C210C4, C40.76(C2×C4), (C8×Dic5)⋊1C2, C10.86(C4×D4), C4.213(D4×D5), C8.C48D5, C52C8.56D4, D207C48C2, D20.25(C2×C4), C20.372(C2×D4), (C2×C8).252D10, D407C2.5C2, (C2×C40).42C22, D20.2C411C2, C20.114(C22×C4), (C2×C20).311C23, Dic10.26(C2×C4), C4○D20.18C22, C2.16(D208C4), C22.2(Q82D5), (C4×Dic5).266C22, (C5×M4(2)).21C22, C4.47(C2×C4×D5), (C5×C8.C4)⋊5C2, (C2×C10).2(C4○D4), (C2×C4).414(C22×D5), (C2×C52C8).246C22, SmallGroup(320,522)

Series: Derived Chief Lower central Upper central

C1C20 — D4013C4
C1C5C10C20C2×C20C4○D20D407C2 — D4013C4
C5C10C20 — D4013C4
C1C4C2×C4C8.C4

Generators and relations for D4013C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a38b >

Subgroups: 390 in 106 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C42, C2×C8, C2×C8 [×3], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×4], C20 [×2], D10 [×2], C2×C10, C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8 [×2], C40 [×2], C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20, C8○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C2×C52C8, C4×Dic5, C2×C40, C5×M4(2) [×2], C4○D20 [×2], C8×Dic5, D207C4 [×2], C5×C8.C4, D407C2, D20.2C4 [×2], D4013C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C8○D8, C2×C4×D5, D4×D5, Q82D5, D208C4, D4013C4

Smallest permutation representation of D4013C4
On 80 points
Generators in S80
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)
(1 11 21 31)(2 20 22 40)(3 29 23 9)(4 38 24 18)(5 7 25 27)(6 16 26 36)(8 34 28 14)(10 12 30 32)(13 39 33 19)(15 17 35 37)(41 77)(42 46)(43 55)(44 64)(45 73)(47 51)(48 60)(49 69)(50 78)(52 56)(53 65)(54 74)(57 61)(58 70)(59 79)(62 66)(63 75)(67 71)(68 80)(72 76)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H8I8J8K8L8M8N10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order1222244444444455888888888888881010101020202020202040···4040···40
size112202011210101010202022222244445555101022442222444···48···8

56 irreducible representations

dim1111111112222222444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5C4○D4D10D10C4×D5C8○D8D4×D5Q82D5D4013C4
kernelD4013C4C8×Dic5D207C4C5×C8.C4D407C2D20.2C4C40⋊C2D40Dic20C52C8C8.C4C2×C10C2×C8M4(2)C8C5C4C22C1
# reps1121124222222488228

Matrix representation of D4013C4 in GL4(𝔽41) generated by

7600
34000
00380
001327
,
7100
343400
00404
0001
,
343500
8700
0090
002340
G:=sub<GL(4,GF(41))| [7,34,0,0,6,0,0,0,0,0,38,13,0,0,0,27],[7,34,0,0,1,34,0,0,0,0,40,0,0,0,4,1],[34,8,0,0,35,7,0,0,0,0,9,23,0,0,0,40] >;

D4013C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{13}C_4
% in TeX

G:=Group("D40:13C4");
// GroupNames label

G:=SmallGroup(320,522);
// by ID

G=gap.SmallGroup(320,522);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,555,58,136,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^38*b>;
// generators/relations

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