G = C10.462+ 1+4 order 320 = 26·5
46th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.462+ 1+4, C4⋊C4⋊8D10, (C2×D4)⋊10D10, C4⋊D4⋊20D5, C22⋊C4⋊12D10, (D4×Dic5)⋊24C2, C23⋊D10⋊13C2, (D4×C10)⋊31C22, C4⋊Dic5⋊12C22, Dic5⋊D4⋊32C2, C20.48D4⋊44C2, (C2×C20).626C23, (C2×C10).161C24, C5⋊5(C22.32C24), (C2×Dic10)⋊8C22, (C4×Dic5)⋊25C22, (C22×C4).228D10, C23.D5⋊26C22, C2.48(D4⋊6D10), C2.29(D4⋊8D10), C23.21(C22×D5), Dic5.5D4⋊22C2, C10.D4⋊34C22, C22.7(D4⋊2D5), C22.D20⋊11C2, (C22×C10).27C23, (C22×D5).68C23, (C23×D5).50C22, C22.182(C23×D5), (C22×C20).312C22, (C2×Dic5).239C23, (C22×Dic5)⋊22C22, D10⋊C4.146C22, C4⋊C4⋊D5⋊14C2, (C5×C4⋊D4)⋊23C2, (C5×C4⋊C4)⋊15C22, C10.85(C2×C4○D4), C2.40(C2×D4⋊2D5), (C2×D10⋊C4)⋊26C2, (C2×C10).23(C4○D4), (C5×C22⋊C4)⋊17C22, (C2×C4).180(C22×D5), (C2×C5⋊D4).34C22, SmallGroup(320,1289)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.462+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=a5, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 1006 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic10, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22.32C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, Dic5.5D4, C22.D20, C4⋊C4⋊D5, C20.48D4, C2×D10⋊C4, D4×Dic5, C23⋊D10, Dic5⋊D4, C5×C4⋊D4, C10.462+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, D4⋊6D10, D4⋊8D10, C10.462+ 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 13 53)(2 44 14 54)(3 45 15 55)(4 46 16 56)(5 47 17 57)(6 48 18 58)(7 49 19 59)(8 50 20 60)(9 41 11 51)(10 42 12 52)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 33 18 28)(2 32 19 27)(3 31 20 26)(4 40 11 25)(5 39 12 24)(6 38 13 23)(7 37 14 22)(8 36 15 21)(9 35 16 30)(10 34 17 29)(41 75 56 70)(42 74 57 69)(43 73 58 68)(44 72 59 67)(45 71 60 66)(46 80 51 65)(47 79 52 64)(48 78 53 63)(49 77 54 62)(50 76 55 61)
(1 28 6 23)(2 29 7 24)(3 30 8 25)(4 21 9 26)(5 22 10 27)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)
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), (1,43,13,53)(2,44,14,54)(3,45,15,55)(4,46,16,56)(5,47,17,57)(6,48,18,58)(7,49,19,59)(8,50,20,60)(9,41,11,51)(10,42,12,52)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,33,18,28)(2,32,19,27)(3,31,20,26)(4,40,11,25)(5,39,12,24)(6,38,13,23)(7,37,14,22)(8,36,15,21)(9,35,16,30)(10,34,17,29)(41,75,56,70)(42,74,57,69)(43,73,58,68)(44,72,59,67)(45,71,60,66)(46,80,51,65)(47,79,52,64)(48,78,53,63)(49,77,54,62)(50,76,55,61), (1,28,6,23)(2,29,7,24)(3,30,8,25)(4,21,9,26)(5,22,10,27)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)>;
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), (1,43,13,53)(2,44,14,54)(3,45,15,55)(4,46,16,56)(5,47,17,57)(6,48,18,58)(7,49,19,59)(8,50,20,60)(9,41,11,51)(10,42,12,52)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,33,18,28)(2,32,19,27)(3,31,20,26)(4,40,11,25)(5,39,12,24)(6,38,13,23)(7,37,14,22)(8,36,15,21)(9,35,16,30)(10,34,17,29)(41,75,56,70)(42,74,57,69)(43,73,58,68)(44,72,59,67)(45,71,60,66)(46,80,51,65)(47,79,52,64)(48,78,53,63)(49,77,54,62)(50,76,55,61), (1,28,6,23)(2,29,7,24)(3,30,8,25)(4,21,9,26)(5,22,10,27)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80) );
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)], [(1,43,13,53),(2,44,14,54),(3,45,15,55),(4,46,16,56),(5,47,17,57),(6,48,18,58),(7,49,19,59),(8,50,20,60),(9,41,11,51),(10,42,12,52),(21,61,31,71),(22,62,32,72),(23,63,33,73),(24,64,34,74),(25,65,35,75),(26,66,36,76),(27,67,37,77),(28,68,38,78),(29,69,39,79),(30,70,40,80)], [(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,33,18,28),(2,32,19,27),(3,31,20,26),(4,40,11,25),(5,39,12,24),(6,38,13,23),(7,37,14,22),(8,36,15,21),(9,35,16,30),(10,34,17,29),(41,75,56,70),(42,74,57,69),(43,73,58,68),(44,72,59,67),(45,71,60,66),(46,80,51,65),(47,79,52,64),(48,78,53,63),(49,77,54,62),(50,76,55,61)], [(1,28,6,23),(2,29,7,24),(3,30,8,25),(4,21,9,26),(5,22,10,27),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,61,46,66),(42,62,47,67),(43,63,48,68),(44,64,49,69),(45,65,50,70),(51,71,56,76),(52,72,57,77),(53,73,58,78),(54,74,59,79),(55,75,60,80)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 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 | 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 | 2 | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊2D5 | D4⋊6D10 | D4⋊8D10 |
| kernel | C10.462+ 1+4 | Dic5.5D4 | C22.D20 | C4⋊C4⋊D5 | C20.48D4 | C2×D10⋊C4 | D4×Dic5 | C23⋊D10 | Dic5⋊D4 | C5×C4⋊D4 | C4⋊D4 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 6 | 2 | 4 | 4 | 4 |
Matrix representation of C10.462+ 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 34 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 27 | 30 |
| 0 | 0 | 0 | 1 | 27 | 27 |
| 0 | 0 | 28 | 19 | 40 | 0 |
| 0 | 0 | 13 | 28 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 19 | 40 | 0 |
| 0 | 0 | 13 | 28 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 14 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 30 | 27 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 14 | 2 | 11 | 32 |
| 0 | 0 | 14 | 14 | 9 | 30 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,1,7,0,0,0,0,34,34],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,28,13,0,0,0,1,19,28,0,0,27,27,40,0,0,0,30,27,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,28,13,0,0,0,1,19,28,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,11,14,0,0,0,0,9,30,0,0,0,0,0,0,14,30,0,0,0,0,14,27],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,11,32,14,14,0,0,9,30,2,14,0,0,0,0,11,9,0,0,0,0,32,30] >;
C10.462+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{46}2_+^{1+4} % in TeX
G:=Group("C10.46ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1289);
// by ID
G=gap.SmallGroup(320,1289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=a^5*b^2,e^2=a^5,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^5*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations