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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, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C22×C10, D4.4D4, C8×D5, C8⋊D5, C4.Dic5, C4.Dic5, D4⋊D5, D4.D5, C2×C40, C5×D8, C4○D20, D4×C10, C40.6C4, C20.D4, D20.3C4, D4.D10, C10×D8, C40.23D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, 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 41 11 71 21 61 31 51)(2 80 12 70 22 60 32 50)(3 79 13 69 23 59 33 49)(4 78 14 68 24 58 34 48)(5 77 15 67 25 57 35 47)(6 76 16 66 26 56 36 46)(7 75 17 65 27 55 37 45)(8 74 18 64 28 54 38 44)(9 73 19 63 29 53 39 43)(10 72 20 62 30 52 40 42)
(1 66)(2 75)(3 44)(4 53)(5 62)(6 71)(7 80)(8 49)(9 58)(10 67)(11 76)(12 45)(13 54)(14 63)(15 72)(16 41)(17 50)(18 59)(19 68)(20 77)(21 46)(22 55)(23 64)(24 73)(25 42)(26 51)(27 60)(28 69)(29 78)(30 47)(31 56)(32 65)(33 74)(34 43)(35 52)(36 61)(37 70)(38 79)(39 48)(40 57)

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

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

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

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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