metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:16D10, C10.182+ 1+4, C4:C4:49D10, (C4xD4):18D5, (D4xC20):20C2, (C22xC4):5D10, (C4xC20):32C22, D10:Q8:8C2, C22:C4:48D10, C4:Dic5:9C22, (C2xD4).217D10, C23:D10.5C2, C42:2D5:16C2, C42:D5:32C2, D10.31(C4oD4), D10.12D4:7C2, C20.48D4:11C2, C23.D5:9C22, (C2xC10).100C24, (C2xC20).699C23, (C22xC20):37C22, Dic5.5D4:7C2, (C2xDic10):6C22, (C4xDic5):52C22, C2.19(D4:6D10), C5:3(C22.45C24), (D4xC10).307C22, C22.12(C4oD20), C10.D4:42C22, (C23xD5).41C22, (C22xD5).35C23, C23.174(C22xD5), C22.125(C23xD5), D10:C4.85C22, C23.11D10:29C2, C23.23D10:16C2, C23.18D10:18C2, (C22xC10).170C23, (C2xDic5).217C23, (C22xDic5).98C22, C4:C4:D5:7C2, (C4xC5:D4):43C2, C2.23(D5xC4oD4), (C5xC4:C4):61C22, (D5xC22:C4):29C2, C2.49(C2xC4oD20), C10.140(C2xC4oD4), (C2xD10:C4):22C2, (C2xC4xD5).252C22, (C2xC10).16(C4oD4), (C5xC22:C4):57C22, (C2xC4).284(C22xD5), (C2xC5:D4).16C22, SmallGroup(320,1228)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:16D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 934 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, 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, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC10, C2xC22:C4, C42:C2, C4xD4, C4xD4, C22wrC2, C22:Q8, C22.D4, C4.4D4, C42:2C2, Dic10, C4xD5, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xC10, C22.45C24, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C5xC4:C4, C2xDic10, C2xC4xD5, C22xDic5, C2xC5:D4, C22xC20, D4xC10, C23xD5, C42:D5, C42:2D5, C23.11D10, D5xC22:C4, D10.12D4, Dic5.5D4, D10:Q8, C4:C4:D5, C20.48D4, C2xD10:C4, C4xC5:D4, C23.23D10, C23.18D10, C23:D10, D4xC20, C42:16D10
Quotients: C1, C2, C22, C23, D5, C4oD4, C24, D10, C2xC4oD4, 2+ 1+4, C22xD5, C22.45C24, C4oD20, C23xD5, C2xC4oD20, D4:6D10, D5xC4oD4, C42:16D10
(1 58 19 66)(2 54 20 62)(3 60 16 68)(4 56 17 64)(5 52 18 70)(6 69 15 51)(7 65 11 57)(8 61 12 53)(9 67 13 59)(10 63 14 55)(21 46 39 79)(22 42 40 75)(23 48 36 71)(24 44 37 77)(25 50 38 73)(26 72 35 49)(27 78 31 45)(28 74 32 41)(29 80 33 47)(30 76 34 43)
(1 28 8 21)(2 29 9 22)(3 30 10 23)(4 26 6 24)(5 27 7 25)(11 38 18 31)(12 39 19 32)(13 40 20 33)(14 36 16 34)(15 37 17 35)(41 53 79 66)(42 54 80 67)(43 55 71 68)(44 56 72 69)(45 57 73 70)(46 58 74 61)(47 59 75 62)(48 60 76 63)(49 51 77 64)(50 52 78 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 2)(3 5)(7 10)(8 9)(11 14)(12 13)(16 18)(19 20)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 75)(42 74)(43 73)(44 72)(45 71)(46 80)(47 79)(48 78)(49 77)(50 76)(51 56)(52 55)(53 54)(57 60)(58 59)(61 62)(63 70)(64 69)(65 68)(66 67)
G:=sub<Sym(80)| (1,58,19,66)(2,54,20,62)(3,60,16,68)(4,56,17,64)(5,52,18,70)(6,69,15,51)(7,65,11,57)(8,61,12,53)(9,67,13,59)(10,63,14,55)(21,46,39,79)(22,42,40,75)(23,48,36,71)(24,44,37,77)(25,50,38,73)(26,72,35,49)(27,78,31,45)(28,74,32,41)(29,80,33,47)(30,76,34,43), (1,28,8,21)(2,29,9,22)(3,30,10,23)(4,26,6,24)(5,27,7,25)(11,38,18,31)(12,39,19,32)(13,40,20,33)(14,36,16,34)(15,37,17,35)(41,53,79,66)(42,54,80,67)(43,55,71,68)(44,56,72,69)(45,57,73,70)(46,58,74,61)(47,59,75,62)(48,60,76,63)(49,51,77,64)(50,52,78,65), (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), (1,2)(3,5)(7,10)(8,9)(11,14)(12,13)(16,18)(19,20)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,56)(52,55)(53,54)(57,60)(58,59)(61,62)(63,70)(64,69)(65,68)(66,67)>;
G:=Group( (1,58,19,66)(2,54,20,62)(3,60,16,68)(4,56,17,64)(5,52,18,70)(6,69,15,51)(7,65,11,57)(8,61,12,53)(9,67,13,59)(10,63,14,55)(21,46,39,79)(22,42,40,75)(23,48,36,71)(24,44,37,77)(25,50,38,73)(26,72,35,49)(27,78,31,45)(28,74,32,41)(29,80,33,47)(30,76,34,43), (1,28,8,21)(2,29,9,22)(3,30,10,23)(4,26,6,24)(5,27,7,25)(11,38,18,31)(12,39,19,32)(13,40,20,33)(14,36,16,34)(15,37,17,35)(41,53,79,66)(42,54,80,67)(43,55,71,68)(44,56,72,69)(45,57,73,70)(46,58,74,61)(47,59,75,62)(48,60,76,63)(49,51,77,64)(50,52,78,65), (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), (1,2)(3,5)(7,10)(8,9)(11,14)(12,13)(16,18)(19,20)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,56)(52,55)(53,54)(57,60)(58,59)(61,62)(63,70)(64,69)(65,68)(66,67) );
G=PermutationGroup([[(1,58,19,66),(2,54,20,62),(3,60,16,68),(4,56,17,64),(5,52,18,70),(6,69,15,51),(7,65,11,57),(8,61,12,53),(9,67,13,59),(10,63,14,55),(21,46,39,79),(22,42,40,75),(23,48,36,71),(24,44,37,77),(25,50,38,73),(26,72,35,49),(27,78,31,45),(28,74,32,41),(29,80,33,47),(30,76,34,43)], [(1,28,8,21),(2,29,9,22),(3,30,10,23),(4,26,6,24),(5,27,7,25),(11,38,18,31),(12,39,19,32),(13,40,20,33),(14,36,16,34),(15,37,17,35),(41,53,79,66),(42,54,80,67),(43,55,71,68),(44,56,72,69),(45,57,73,70),(46,58,74,61),(47,59,75,62),(48,60,76,63),(49,51,77,64),(50,52,78,65)], [(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)], [(1,2),(3,5),(7,10),(8,9),(11,14),(12,13),(16,18),(19,20),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,75),(42,74),(43,73),(44,72),(45,71),(46,80),(47,79),(48,78),(49,77),(50,76),(51,56),(52,55),(53,54),(57,60),(58,59),(61,62),(63,70),(64,69),(65,68),(66,67)]])
65 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4oD4 | C4oD4 | D10 | D10 | D10 | D10 | D10 | C4oD20 | 2+ 1+4 | D4:6D10 | D5xC4oD4 |
kernel | C42:16D10 | C42:D5 | C42:2D5 | C23.11D10 | D5xC22:C4 | D10.12D4 | Dic5.5D4 | D10:Q8 | C4:C4:D5 | C20.48D4 | C2xD10:C4 | C4xC5:D4 | C23.23D10 | C23.18D10 | C23:D10 | D4xC20 | C4xD4 | D10 | C2xC10 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | C22 | C10 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42:16D10 ►in GL6(F41)
40 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 9 | 0 | 0 |
0 | 0 | 0 | 0 | 14 | 18 |
0 | 0 | 0 | 0 | 37 | 27 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 15 | 37 | 0 | 0 |
0 | 0 | 36 | 26 | 0 | 0 |
0 | 0 | 0 | 0 | 9 | 0 |
0 | 0 | 0 | 0 | 0 | 9 |
0 | 7 | 0 | 0 | 0 | 0 |
35 | 6 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 3 | 40 |
35 | 7 | 0 | 0 | 0 | 0 |
36 | 6 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 28 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 3 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,37,0,0,0,0,18,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,36,0,0,0,0,37,26,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,35,0,0,0,0,7,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[35,36,0,0,0,0,7,6,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,3,0,0,0,0,0,40] >;
C42:16D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{16}D_{10}
% in TeX
G:=Group("C4^2:16D10");
// GroupNames label
G:=SmallGroup(320,1228);
// by ID
G=gap.SmallGroup(320,1228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations