G = C10.562+ 1+4 order 320 = 26·5
56th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.562+ 1+4, C4⋊C4⋊13D10, (C2×Q8)⋊7D10, C22⋊Q8⋊21D5, C20⋊7D4⋊46C2, C4⋊D20⋊27C2, (C2×D20)⋊8C22, C22⋊D20⋊17C2, (C2×C20).64C23, C4⋊Dic5⋊14C22, C22⋊C4.64D10, (Q8×C10)⋊10C22, Dic5⋊4D4⋊15C2, C20.23D4⋊17C2, (C2×C10).188C24, C5⋊6(C22.32C24), (C4×Dic5)⋊30C22, (C22×C4).250D10, D10.13D4⋊22C2, C2.58(D4⋊6D10), C2.38(D4⋊8D10), D10⋊C4⋊33C22, C10.D4⋊20C22, C22.4(Q8⋊2D5), (C22×D5).79C23, (C23×D5).55C22, C22.209(C23×D5), C23.196(C22×D5), (C22×C10).216C23, (C22×C20).316C22, (C2×Dic5).251C23, (C22×Dic5).124C22, (C2×C4×D5)⋊19C22, C4⋊C4⋊D5⋊23C2, (C5×C4⋊C4)⋊22C22, (C5×C22⋊Q8)⋊24C2, C10.116(C2×C4○D4), C2.20(C2×Q8⋊2D5), (C2×D10⋊C4)⋊27C2, (C2×C10).28(C4○D4), (C2×C4).185(C22×D5), (C2×C5⋊D4).40C22, (C5×C22⋊C4).43C22, SmallGroup(320,1316)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.562+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a5b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=a5b2d >
Subgroups: 1102 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, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, C22.32C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, Q8×C10, C23×D5, Dic5⋊4D4, C22⋊D20, D10.13D4, C4⋊D20, C4⋊C4⋊D5, C2×D10⋊C4, C20⋊7D4, C20.23D4, C5×C22⋊Q8, C10.562+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, Q8⋊2D5, C23×D5, D4⋊6D10, C2×Q8⋊2D5, D4⋊8D10, C10.562+ 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 48 18 53)(2 49 19 54)(3 50 20 55)(4 41 11 56)(5 42 12 57)(6 43 13 58)(7 44 14 59)(8 45 15 60)(9 46 16 51)(10 47 17 52)(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)
(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 38 18 23)(2 37 19 22)(3 36 20 21)(4 35 11 30)(5 34 12 29)(6 33 13 28)(7 32 14 27)(8 31 15 26)(9 40 16 25)(10 39 17 24)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)(46 75 51 70)(47 74 52 69)(48 73 53 68)(49 72 54 67)(50 71 55 66)
(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), (1,48,18,53)(2,49,19,54)(3,50,20,55)(4,41,11,56)(5,42,12,57)(6,43,13,58)(7,44,14,59)(8,45,15,60)(9,46,16,51)(10,47,17,52)(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), (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), (1,38,18,23)(2,37,19,22)(3,36,20,21)(4,35,11,30)(5,34,12,29)(6,33,13,28)(7,32,14,27)(8,31,15,26)(9,40,16,25)(10,39,17,24)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)(46,75,51,70)(47,74,52,69)(48,73,53,68)(49,72,54,67)(50,71,55,66), (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), (1,48,18,53)(2,49,19,54)(3,50,20,55)(4,41,11,56)(5,42,12,57)(6,43,13,58)(7,44,14,59)(8,45,15,60)(9,46,16,51)(10,47,17,52)(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), (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), (1,38,18,23)(2,37,19,22)(3,36,20,21)(4,35,11,30)(5,34,12,29)(6,33,13,28)(7,32,14,27)(8,31,15,26)(9,40,16,25)(10,39,17,24)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)(46,75,51,70)(47,74,52,69)(48,73,53,68)(49,72,54,67)(50,71,55,66), (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)], [(1,48,18,53),(2,49,19,54),(3,50,20,55),(4,41,11,56),(5,42,12,57),(6,43,13,58),(7,44,14,59),(8,45,15,60),(9,46,16,51),(10,47,17,52),(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)], [(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)], [(1,38,18,23),(2,37,19,22),(3,36,20,21),(4,35,11,30),(5,34,12,29),(6,33,13,28),(7,32,14,27),(8,31,15,26),(9,40,16,25),(10,39,17,24),(41,80,56,65),(42,79,57,64),(43,78,58,63),(44,77,59,62),(45,76,60,61),(46,75,51,70),(47,74,52,69),(48,73,53,68),(49,72,54,67),(50,71,55,66)], [(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 | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| 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 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 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 | Q8⋊2D5 | D4⋊6D10 | D4⋊8D10 |
| kernel | C10.562+ 1+4 | Dic5⋊4D4 | C22⋊D20 | D10.13D4 | C4⋊D20 | C4⋊C4⋊D5 | C2×D10⋊C4 | C20⋊7D4 | C20.23D4 | C5×C22⋊Q8 | C22⋊Q8 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 6 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C10.562+ 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 35 |
| 0 | 0 | 0 | 0 | 6 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 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 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 18 | 0 | 0 |
| 0 | 0 | 21 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 18 |
| 0 | 0 | 0 | 0 | 21 | 20 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 6 | 0 | 0 |
| 0 | 0 | 35 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 6 |
| 0 | 0 | 0 | 0 | 35 | 18 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,21,21,0,0,0,0,18,20,0,0,0,0,0,0,21,21,0,0,0,0,18,20],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,23,35,0,0,0,0,6,18,0,0,0,0,0,0,23,35,0,0,0,0,6,18] >;
C10.562+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{56}2_+^{1+4} % in TeX
G:=Group("C10.56ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1316);
// by ID
G=gap.SmallGroup(320,1316);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,184,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=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=a^5*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=a^5*b^2*d>;
// generators/relations