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
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
(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 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4oD4 | C4oD4 | D10 | D10 | D10 | D10 | D10 | C4oD20 | D4xD5 | D5xC4oD4 |
kernel | C42:12D10 | C42:D5 | C4xD20 | Dic5.14D4 | C22:D20 | D10.12D4 | D10:D4 | D10.13D4 | D10:Q8 | C4xC5:D4 | C23:D10 | Dic5:D4 | D4xC20 | D5xC22xC4 | C2xC4oD20 | C4xD5 | C4xD4 | D10 | C2xC10 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | C22 | C4 | C2 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 4 | 4 |
Matrix representation of C42:12D10 ►in GL4(F41) generated by
1 | 40 | 0 | 0 |
2 | 40 | 0 | 0 |
0 | 0 | 18 | 35 |
0 | 0 | 6 | 23 |
9 | 0 | 0 | 0 |
0 | 9 | 0 | 0 |
0 | 0 | 9 | 0 |
0 | 0 | 0 | 9 |
40 | 1 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 6 | 6 |
0 | 0 | 35 | 1 |
40 | 0 | 0 | 0 |
0 | 40 | 0 | 0 |
0 | 0 | 6 | 6 |
0 | 0 | 1 | 35 |
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