G = C10.512+ 1+4 order 320 = 26·5
51st non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.512+ 1+4, C4⋊C4⋊11D10, (C2×Q8)⋊5D10, C22⋊Q8⋊11D5, D10.2(C2×Q8), (C22×D5)⋊3Q8, C22.7(Q8×D5), (Q8×C10)⋊8C22, C5⋊2(C23⋊2Q8), D10⋊3Q8⋊16C2, D10⋊Q8⋊21C2, D10⋊2Q8⋊27C2, (C2×C20).57C23, C4⋊Dic5⋊36C22, C22⋊C4.59D10, C20.48D4⋊46C2, C10.36(C22×Q8), (C2×C10).178C24, (C2×Dic10)⋊9C22, (C22×C4).240D10, C2.53(D4⋊6D10), C2.35(D4⋊8D10), C10.D4⋊18C22, (C2×Dic5).89C23, (C23×D5).53C22, C23.191(C22×D5), C22.199(C23×D5), Dic5.14D4⋊24C2, C23.D5.34C22, (C22×C10).206C23, (C22×C20).315C22, (C22×D5).210C23, D10⋊C4.147C22, (C22×Dic5).119C22, C2.19(C2×Q8×D5), (C2×C10).7(C2×Q8), (C5×C4⋊C4)⋊20C22, (C5×C22⋊Q8)⋊14C2, (D5×C22⋊C4).2C2, (C2×C4×D5).107C22, (C2×C4).183(C22×D5), (C2×D10⋊C4).22C2, (C5×C22⋊C4).33C22, SmallGroup(320,1306)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.512+ 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, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=a5b2d >
Subgroups: 958 in 242 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C22⋊Q8, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, C23⋊2Q8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C22×Dic5, C22×C20, Q8×C10, C23×D5, Dic5.14D4, D5×C22⋊C4, D10⋊Q8, D10⋊2Q8, C20.48D4, C2×D10⋊C4, D10⋊3Q8, C5×C22⋊Q8, C10.512+ 1+4
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, C22×D5, C23⋊2Q8, Q8×D5, C23×D5, D4⋊6D10, C2×Q8×D5, D4⋊8D10, C10.512+ 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 61 36 76)(22 62 37 77)(23 63 38 78)(24 64 39 79)(25 65 40 80)(26 66 31 71)(27 67 32 72)(28 68 33 73)(29 69 34 74)(30 70 35 75)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(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 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), (1,43,18,58)(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)(21,61,36,76)(22,62,37,77)(23,63,38,78)(24,64,39,79)(25,65,40,80)(26,66,31,71)(27,67,32,72)(28,68,33,73)(29,69,34,74)(30,70,35,75), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (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)(41,80,56,65)(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), (1,43,18,58)(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)(21,61,36,76)(22,62,37,77)(23,63,38,78)(24,64,39,79)(25,65,40,80)(26,66,31,71)(27,67,32,72)(28,68,33,73)(29,69,34,74)(30,70,35,75), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (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)(41,80,56,65)(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)], [(1,43,18,58),(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),(21,61,36,76),(22,62,37,77),(23,63,38,78),(24,64,39,79),(25,65,40,80),(26,66,31,71),(27,67,32,72),(28,68,33,73),(29,69,34,74),(30,70,35,75)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)], [(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),(41,80,56,65),(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 | ··· | 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 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | ··· | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | Q8×D5 | D4⋊6D10 | D4⋊8D10 |
| kernel | C10.512+ 1+4 | Dic5.14D4 | D5×C22⋊C4 | D10⋊Q8 | D10⋊2Q8 | C20.48D4 | C2×D10⋊C4 | D10⋊3Q8 | C5×C22⋊Q8 | C22×D5 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 6 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C10.512+ 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 | 34 | 34 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 26 | 4 | 0 | 0 | 0 | 0 |
| 5 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 14 | 30 |
| 26 | 4 | 0 | 0 | 0 | 0 |
| 5 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[12,12,0,0,0,0,5,29,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,5,0,0,0,0,4,15,0,0,0,0,0,0,30,27,0,0,0,0,32,11,0,0,0,0,0,0,11,14,0,0,0,0,9,30],[26,5,0,0,0,0,4,15,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,11,32,0,0,0,0,9,30] >;
C10.512+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{51}2_+^{1+4} % in TeX
G:=Group("C10.51ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1306);
// by ID
G=gap.SmallGroup(320,1306);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,297,136,12550]);
// 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,d*b*d^-1=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