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G = D4016C4order 320 = 26·5

10th semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4016C4, Dic2016C4, M4(2).26D10, C40⋊C28C4, C408C41C2, C8.11(C4×D5), C56(C8.26D4), C40.70(C2×C4), (C2×C8).69D10, C4.212(D4×D5), C10.85(C4×D4), C8.C44D5, C52C8.51D4, D207C47C2, D20.24(C2×C4), C20.371(C2×D4), D407C2.4C2, (C2×C40).41C22, D20.2C410C2, (C2×C20).310C23, C20.113(C22×C4), Dic10.25(C2×C4), C4○D20.17C22, C2.15(D208C4), C22.1(Q82D5), (C4×Dic5).46C22, (C5×M4(2)).20C22, C4.46(C2×C4×D5), (C5×C8.C4)⋊4C2, (C2×C10).1(C4○D4), (C2×C52C8).78C22, (C2×C4).413(C22×D5), SmallGroup(320,521)

Series: Derived Chief Lower central Upper central

C1C20 — D4016C4
C1C5C10C20C2×C20C4○D20D407C2 — D4016C4
C5C10C20 — D4016C4
C1C4C2×C4C8.C4

Generators and relations for D4016C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a18b >

Subgroups: 390 in 104 conjugacy classes, 45 normal (27 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], 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 [×3], C20 [×2], D10 [×2], C2×C10, C8⋊C4, 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, C8.26D4, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C2×C52C8, C4×Dic5, C2×C40, C5×M4(2) [×2], C4○D20 [×2], C408C4, D207C4 [×2], C5×C8.C4, D407C2, D20.2C4 [×2], D4016C4
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, C8.26D4, C2×C4×D5, D4×D5, Q82D5, D208C4, D4016C4

Smallest permutation representation of D4016C4
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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 80)(38 79)(39 78)(40 77)
(1 11 21 31)(2 40 22 20)(3 29 23 9)(4 18 24 38)(5 7 25 27)(6 36 26 16)(8 14 28 34)(10 32 30 12)(13 39 33 19)(15 17 35 37)(41 53)(43 71)(44 60)(45 49)(46 78)(47 67)(48 56)(50 74)(51 63)(54 70)(55 59)(57 77)(58 66)(61 73)(64 80)(65 69)(68 76)(75 79)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A···8F8G8H8I8J10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order122224444444558···888881010101020202020202040···4040···40
size112202011220202020224···41010101022442222444···48···8

50 irreducible representations

dim1111111112222224444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5C4○D4D10D10C4×D5C8.26D4D4×D5Q82D5D4016C4
kernelD4016C4C408C4D207C4C5×C8.C4D407C2D20.2C4C40⋊C2D40Dic20C52C8C8.C4C2×C10C2×C8M4(2)C8C5C4C22C1
# reps1121124222222482228

Matrix representation of D4016C4 in GL6(𝔽41)

660000
3510000
0031010
00174001
00140100
00238241
,
660000
1350000
0004000
0040000
00302209
00717320
,
900000
13320000
0032000
000100
0025090
00132040

G:=sub<GL(6,GF(41))| [6,35,0,0,0,0,6,1,0,0,0,0,0,0,31,17,14,23,0,0,0,40,0,8,0,0,1,0,10,24,0,0,0,1,0,1],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,0,40,30,7,0,0,40,0,22,17,0,0,0,0,0,32,0,0,0,0,9,0],[9,13,0,0,0,0,0,32,0,0,0,0,0,0,32,0,25,13,0,0,0,1,0,2,0,0,0,0,9,0,0,0,0,0,0,40] >;

D4016C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{16}C_4
% in TeX

G:=Group("D40:16C4");
// GroupNames label

G:=SmallGroup(320,521);
// by ID

G=gap.SmallGroup(320,521);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,136,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^18*b>;
// generators/relations

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