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G = C40.23D4order 320 = 26·5

23rd non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.23D4, D20.25D4, Dic10.25D4, (C10×D8)⋊2C2, (C2×D8)⋊10D5, C4.61(D4×D5), (C2×C8).87D10, C40.6C43C2, (C2×D4).67D10, C20.169(C2×D4), C20.D47C2, C54(D4.4D4), C8.27(C5⋊D4), D4.D104C2, D20.3C41C2, (C2×C40).32C22, C2.18(C202D4), (C2×C20).437C23, C4○D20.46C22, (D4×C10).86C22, C10.111(C4⋊D4), C4.Dic5.19C22, C22.20(D42D5), C4.81(C2×C5⋊D4), (C2×C4).126(C22×D5), (C2×C10).158(C4○D4), SmallGroup(320,787)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.23D4
C1C5C10C20C2×C20C4○D20D20.3C4 — C40.23D4
C5C10C2×C20 — C40.23D4
C1C2C2×C4C2×D8

Generators and relations for C40.23D4
 G = < a,b,c | a40=c2=1, b4=a20, bab-1=a-1, cac=a9, cbc=a20b3 >

Subgroups: 398 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C8 [×3], C2×C4, C2×C4, D4 [×6], Q8, C23 [×2], D5, C10, C10 [×3], C2×C8, C2×C8, M4(2) [×4], D8 [×4], SD16 [×2], C2×D4 [×2], C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], C52C8 [×3], C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×4], C22×C10 [×2], D4.4D4, C8×D5, C8⋊D5, C4.Dic5, C4.Dic5 [×2], D4⋊D5 [×2], D4.D5 [×2], C2×C40, C5×D8 [×2], C4○D20, D4×C10 [×2], C40.6C4, C20.D4 [×2], D20.3C4, D4.D10 [×2], C10×D8, C40.23D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4.4D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, C40.23D4

Smallest permutation representation of C40.23D4
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 51 11 41 21 71 31 61)(2 50 12 80 22 70 32 60)(3 49 13 79 23 69 33 59)(4 48 14 78 24 68 34 58)(5 47 15 77 25 67 35 57)(6 46 16 76 26 66 36 56)(7 45 17 75 27 65 37 55)(8 44 18 74 28 64 38 54)(9 43 19 73 29 63 39 53)(10 42 20 72 30 62 40 52)
(1 76)(2 45)(3 54)(4 63)(5 72)(6 41)(7 50)(8 59)(9 68)(10 77)(11 46)(12 55)(13 64)(14 73)(15 42)(16 51)(17 60)(18 69)(19 78)(20 47)(21 56)(22 65)(23 74)(24 43)(25 52)(26 61)(27 70)(28 79)(29 48)(30 57)(31 66)(32 75)(33 44)(34 53)(35 62)(36 71)(37 80)(38 49)(39 58)(40 67)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B8C8D8E8F8G10A···10F10G···10N20A20B20C20D40A···40H
order12222244455888888810···1010···102020202040···40
size1128820222022224202040402···28···844444···4

44 irreducible representations

dim111111222222224444
type++++++++++++++-
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10C5⋊D4D4.4D4D4×D5D42D5C40.23D4
kernelC40.23D4C40.6C4C20.D4D20.3C4D4.D10C10×D8C40Dic10D20C2×D8C2×C10C2×C8C2×D4C8C5C4C22C1
# reps112121211222482228

Matrix representation of C40.23D4 in GL6(𝔽41)

1600000
0180000
0004000
0012400
0000040
0000124
,
0150000
3000000
0000040
0000400
00174000
0012400
,
0150000
1100000
000010
000001
001000
000100

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,40,24,0,0,0,0,0,0,0,1,0,0,0,0,40,24],[0,30,0,0,0,0,15,0,0,0,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,40,0,0,0,0,40,0,0,0],[0,11,0,0,0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C40.23D4 in GAP, Magma, Sage, TeX

C_{40}._{23}D_4
% in TeX

G:=Group("C40.23D4");
// GroupNames label

G:=SmallGroup(320,787);
// by ID

G=gap.SmallGroup(320,787);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,1123,297,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations

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