G = C10.422+ 1+4 order 320 = 26·5
42nd non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.422+ 1+4, C4⋊C4⋊7D10, (C2×D4)⋊9D10, C4⋊D4⋊16D5, C20⋊2D4⋊22C2, C22⋊C4⋊29D10, (C22×C4)⋊20D10, C23⋊D10⋊12C2, D10⋊Q8⋊15C2, (D4×C10)⋊15C22, (C2×C20).42C23, C4⋊Dic5⋊33C22, D10.10(C4○D4), Dic5⋊D4⋊14C2, Dic5⋊4D4⋊10C2, C20.17D4⋊18C2, (C2×C10).157C24, (C22×C20)⋊41C22, C5⋊4(C22.32C24), (C4×Dic5)⋊24C22, C23.D5⋊25C22, C2.44(D4⋊6D10), C23.17(C22×D5), Dic5.5D4⋊20C2, (C2×Dic10)⋊26C22, C10.D4⋊29C22, C23.D10⋊18C2, (C22×C10).24C23, (C2×Dic5).76C23, (C23×D5).49C22, (C22×D5).65C23, C22.178(C23×D5), D10⋊C4.70C22, C23.23D10⋊22C2, C23.18D10⋊22C2, (C22×Dic5)⋊21C22, (C4×C5⋊D4)⋊55C2, (D5×C22⋊C4)⋊7C2, C2.41(D5×C4○D4), C4⋊C4⋊D5⋊13C2, (C5×C4⋊D4)⋊19C2, (C5×C4⋊C4)⋊14C22, (C2×C4×D5).94C22, C10.154(C2×C4○D4), (C5×C22⋊C4)⋊16C22, (C2×C4).178(C22×D5), (C2×C5⋊D4).30C22, SmallGroup(320,1285)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.422+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=a5b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a5b-1, bd=db, be=eb, dcd-1=ece=a5c, ede=a5b2d >
Subgroups: 1006 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22.32C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C23.D10, D5×C22⋊C4, Dic5⋊4D4, Dic5.5D4, D10⋊Q8, C4⋊C4⋊D5, C4×C5⋊D4, C23.23D10, C23.18D10, C20.17D4, C23⋊D10, C20⋊2D4, Dic5⋊D4, C5×C4⋊D4, C10.422+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, C23×D5, D4⋊6D10, D5×C4○D4, C10.422+ 1+4
►On 80 points
(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 43 18 58)(2 44 19 59)(3 45 20 60)(4 46 11 51)(5 47 12 52)(6 48 13 53)(7 49 14 54)(8 50 15 55)(9 41 16 56)(10 42 17 57)(21 66 36 71)(22 67 37 72)(23 68 38 73)(24 69 39 74)(25 70 40 75)(26 61 31 76)(27 62 32 77)(28 63 33 78)(29 64 34 79)(30 65 35 80)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 33 13 23)(2 32 14 22)(3 31 15 21)(4 40 16 30)(5 39 17 29)(6 38 18 28)(7 37 19 27)(8 36 20 26)(9 35 11 25)(10 34 12 24)(41 80 51 70)(42 79 52 69)(43 78 53 68)(44 77 54 67)(45 76 55 66)(46 75 56 65)(47 74 57 64)(48 73 58 63)(49 72 59 62)(50 71 60 61)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
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), (1,43,18,58)(2,44,19,59)(3,45,20,60)(4,46,11,51)(5,47,12,52)(6,48,13,53)(7,49,14,54)(8,50,15,55)(9,41,16,56)(10,42,17,57)(21,66,36,71)(22,67,37,72)(23,68,38,73)(24,69,39,74)(25,70,40,75)(26,61,31,76)(27,62,32,77)(28,63,33,78)(29,64,34,79)(30,65,35,80), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,33,13,23)(2,32,14,22)(3,31,15,21)(4,40,16,30)(5,39,17,29)(6,38,18,28)(7,37,19,27)(8,36,20,26)(9,35,11,25)(10,34,12,24)(41,80,51,70)(42,79,52,69)(43,78,53,68)(44,77,54,67)(45,76,55,66)(46,75,56,65)(47,74,57,64)(48,73,58,63)(49,72,59,62)(50,71,60,61), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)>;
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), (1,43,18,58)(2,44,19,59)(3,45,20,60)(4,46,11,51)(5,47,12,52)(6,48,13,53)(7,49,14,54)(8,50,15,55)(9,41,16,56)(10,42,17,57)(21,66,36,71)(22,67,37,72)(23,68,38,73)(24,69,39,74)(25,70,40,75)(26,61,31,76)(27,62,32,77)(28,63,33,78)(29,64,34,79)(30,65,35,80), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,33,13,23)(2,32,14,22)(3,31,15,21)(4,40,16,30)(5,39,17,29)(6,38,18,28)(7,37,19,27)(8,36,20,26)(9,35,11,25)(10,34,12,24)(41,80,51,70)(42,79,52,69)(43,78,53,68)(44,77,54,67)(45,76,55,66)(46,75,56,65)(47,74,57,64)(48,73,58,63)(49,72,59,62)(50,71,60,61), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75) );
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)], [(1,43,18,58),(2,44,19,59),(3,45,20,60),(4,46,11,51),(5,47,12,52),(6,48,13,53),(7,49,14,54),(8,50,15,55),(9,41,16,56),(10,42,17,57),(21,66,36,71),(22,67,37,72),(23,68,38,73),(24,69,39,74),(25,70,40,75),(26,61,31,76),(27,62,32,77),(28,63,33,78),(29,64,34,79),(30,65,35,80)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,33,13,23),(2,32,14,22),(3,31,15,21),(4,40,16,30),(5,39,17,29),(6,38,18,28),(7,37,19,27),(8,36,20,26),(9,35,11,25),(10,34,12,24),(41,80,51,70),(42,79,52,69),(43,78,53,68),(44,77,54,67),(45,76,55,66),(46,75,56,65),(47,74,57,64),(48,73,58,63),(49,72,59,62),(50,71,60,61)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 |
| kernel | C10.422+ 1+4 | C23.D10 | D5×C22⋊C4 | Dic5⋊4D4 | Dic5.5D4 | D10⋊Q8 | C4⋊C4⋊D5 | C4×C5⋊D4 | C23.23D10 | C23.18D10 | C20.17D4 | C23⋊D10 | C20⋊2D4 | Dic5⋊D4 | C5×C4⋊D4 | C4⋊D4 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 6 | 2 | 8 | 4 |
Matrix representation of C10.422+ 1+4 ►in GL8(𝔽41)
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 34 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 34 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 2 | 40 | 0 |
| 0 | 0 | 0 | 0 | 39 | 7 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 | 40 | 24 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 27 | 17 | 3 |
| 0 | 0 | 0 | 0 | 0 | 39 | 40 | 24 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 0 | 39 | 40 | 24 |
G:=sub<GL(8,GF(41))| [6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,34,34],[9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,34,39,0,0,0,0,0,1,2,7,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,17,40,1,0,0,0,0,0,1,24,0,1],[0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,17,40,2,0,0,0,0,0,3,24,27,39,0,0,0,0,0,0,17,40,0,0,0,0,0,0,3,24],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,1,39,0,0,0,0,0,40,17,0,39,0,0,0,0,0,0,17,40,0,0,0,0,0,0,1,24] >;
C10.422+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{42}2_+^{1+4} % in TeX
G:=Group("C10.42ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1285);
// by ID
G=gap.SmallGroup(320,1285);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=a^5*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^5*b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*b^2*d>;
// generators/relations