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G = C42:12D10order 320 = 26·5

12nd semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:12D10, (C4xD4):9D5, C4:C4:45D10, (C4xD5):14D4, (C4xD20):25C2, (D4xC20):11C2, C4.219(D4xD5), D10:2(C4oD4), (C4xC20):17C22, C22:C4:44D10, D10.72(C2xD4), C20.378(C2xD4), (C22xC4):10D10, C22:D20:30C2, D10:D4:47C2, C23:D10:33C2, D10:Q8:51C2, (C2xD4).211D10, C22:2(C4oD20), C42:D5:13C2, (C2xC10).91C24, C4:Dic5:57C22, Dic5.83(C2xD4), C10.47(C22xD4), Dic5:D4:46C2, (C2xC20).157C23, (C22xC20):15C22, C5:2(C22.19C24), (C4xDic5):51C22, D10.13D4:49C2, D10.12D4:53C2, C23.D5:49C22, D10:C4:65C22, (C2xDic10):52C22, (C2xD20).217C22, (D4xC10).304C22, C10.D4:70C22, (C2xDic5).38C23, C22.116(C23xD5), C23.170(C22xD5), Dic5.14D4:50C2, (C22xC10).161C23, (C22xD5).180C23, (C23xD5).118C22, (C22xDic5).243C22, C2.19(C2xD4xD5), (C2xC4oD20):5C2, (C4xC5:D4):42C2, C2.20(D5xC4oD4), (C2xC4xD5):47C22, (D5xC22xC4):22C2, (C2xC10):1(C4oD4), (C5xC4:C4):57C22, C10.39(C2xC4oD4), C2.43(C2xC4oD20), (C2xC5:D4):37C22, (C5xC22:C4):55C22, (C2xC4).156(C22xD5), SmallGroup(320,1219)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C42:12D10
C1C5C10C2xC10C22xD5C23xD5D5xC22xC4 — C42:12D10
C5C2xC10 — C42:12D10
C1C2xC4C4xD4

Generators and relations for C42:12D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1246 in 330 conjugacy classes, 109 normal (91 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC10, C42:C2, C4xD4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C23xC4, C2xC4oD4, Dic10, C4xD5, C4xD5, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xC10, C22.19C24, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C5xC4:C4, C2xDic10, C2xC4xD5, C2xC4xD5, C2xD20, C4oD20, C22xDic5, C2xC5:D4, C22xC20, D4xC10, C23xD5, C42:D5, C4xD20, Dic5.14D4, C22:D20, D10.12D4, D10:D4, D10.13D4, D10:Q8, C4xC5:D4, C23:D10, Dic5:D4, D4xC20, D5xC22xC4, C2xC4oD20, C42:12D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, C24, D10, C22xD4, C2xC4oD4, C22xD5, C22.19C24, C4oD20, D4xD5, C23xD5, C2xC4oD20, C2xD4xD5, D5xC4oD4, C42:12D10

Smallest permutation representation of C42:12D10
On 80 points
Generators in S80
(1 33 25 45)(2 46 26 34)(3 35 27 47)(4 48 28 36)(5 37 29 49)(6 50 30 38)(7 39 21 41)(8 42 22 40)(9 31 23 43)(10 44 24 32)(11 52 73 65)(12 66 74 53)(13 54 75 67)(14 68 76 55)(15 56 77 69)(16 70 78 57)(17 58 79 61)(18 62 80 59)(19 60 71 63)(20 64 72 51)
(1 57 6 52)(2 58 7 53)(3 59 8 54)(4 60 9 55)(5 51 10 56)(11 45 78 38)(12 46 79 39)(13 47 80 40)(14 48 71 31)(15 49 72 32)(16 50 73 33)(17 41 74 34)(18 42 75 35)(19 43 76 36)(20 44 77 37)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(21 28)(22 27)(23 26)(24 25)(29 30)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 52)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 70)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10F10G···10N20A···20H20I···20X
order12222222222244444444444444445510···1010···1020···2020···20
size1111224101010102011112244410101010202020222···24···42···24···4

68 irreducible representations

dim111111111111111222222222244
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D5C4oD4C4oD4D10D10D10D10D10C4oD20D4xD5D5xC4oD4
kernelC42:12D10C42:D5C4xD20Dic5.14D4C22:D20D10.12D4D10:D4D10.13D4D10:Q8C4xC5:D4C23:D10Dic5:D4D4xC20D5xC22xC4C2xC4oD20C4xD5C4xD4D10C2xC10C42C22:C4C4:C4C22xC4C2xD4C22C4C2
# reps1111111112111114244242421644

Matrix representation of C42:12D10 in GL4(F41) generated by

14000
24000
001835
00623
,
9000
0900
0090
0009
,
40100
0100
0066
00351
,
40000
04000
0066
00135
G:=sub<GL(4,GF(41))| [1,2,0,0,40,40,0,0,0,0,18,6,0,0,35,23],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[40,0,0,0,1,1,0,0,0,0,6,35,0,0,6,1],[40,0,0,0,0,40,0,0,0,0,6,1,0,0,6,35] >;

C42:12D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{12}D_{10}
% in TeX

G:=Group("C4^2:12D10");
// GroupNames label

G:=SmallGroup(320,1219);
// by ID

G=gap.SmallGroup(320,1219);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,675,570,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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x
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Z
F
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