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G = C40.44D4order 320 = 26·5

44th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.44D4, D20.26D4, Dic10.26D4, C4.66(D4×D5), (C2×SD16)⋊3D5, (C2×C8).92D10, (C10×SD16)⋊1C2, (C2×D4).78D10, C20.D48C2, C20.180(C2×D4), C55(D4.3D4), C8.32(C5⋊D4), (C2×Q8).59D10, C40.6C410C2, D20.3C44C2, C20.C234C2, C20.10D47C2, (C2×C40).48C22, C2.21(C202D4), (C2×C20).454C23, D4.D10.2C2, C4○D20.47C22, (Q8×C10).83C22, C10.118(C4⋊D4), (D4×C10).103C22, C4.Dic5.20C22, C22.21(D42D5), C4.84(C2×C5⋊D4), (C2×C4).127(C22×D5), (C2×C10).159(C4○D4), SmallGroup(320,804)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.44D4
C1C5C10C20C2×C20C4○D20D20.3C4 — C40.44D4
C5C10C2×C20 — C40.44D4
C1C2C2×C4C2×SD16

Generators and relations for C40.44D4
 G = < a,b,c | a40=c2=1, b4=a20, bab-1=a19, cac=a9, cbc=a20b3 >

Subgroups: 366 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, D5, C10, C10 [×2], C2×C8, C2×C8, M4(2) [×4], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20 [×2], C20, D10, C2×C10, C2×C10 [×2], C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8 [×3], C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×C10, D4.3D4, C8×D5, C8⋊D5, C4.Dic5 [×3], D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C5×SD16 [×2], C4○D20, D4×C10, Q8×C10, C40.6C4, C20.D4, C20.10D4, D20.3C4, D4.D10, C20.C23, C10×SD16, C40.44D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4.3D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, C40.44D4

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222444455888888810···1010101010202020202020202040···40
size1128202282022224202040402···28888444488884···4

44 irreducible representations

dim111111112222222224444
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4D4.3D4D4×D5D42D5C40.44D4
kernelC40.44D4C40.6C4C20.D4C20.10D4D20.3C4D4.D10C20.C23C10×SD16C40Dic10D20C2×SD16C2×C10C2×C8C2×D4C2×Q8C8C5C4C22C1
# reps111111112112222282228

Matrix representation of C40.44D4 in GL4(𝔽41) generated by

0253030
40383021
00023
004028
,
34171627
389332
13391616
0383423
,
738316
318319
2803413
0383423
G:=sub<GL(4,GF(41))| [0,40,0,0,25,38,0,0,30,30,0,40,30,21,23,28],[34,38,13,0,17,9,39,38,16,33,16,34,27,2,16,23],[7,3,28,0,38,18,0,38,3,31,34,34,16,9,13,23] >;

C40.44D4 in GAP, Magma, Sage, TeX

C_{40}._{44}D_4
% in TeX

G:=Group("C40.44D4");
// GroupNames label

G:=SmallGroup(320,804);
// by ID

G=gap.SmallGroup(320,804);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,1123,297,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^19,c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations

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