?
G = C10.1462+ (1+4) order 320 = 26·5
55th non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.1462+ (1+4), (C2×C20)⋊16D4, (C2×D4)⋊45D10, (C2×Q8)⋊34D10, C20⋊7D4⋊48C2, C20⋊D4⋊30C2, C20.430(C2×D4), (C22×C4)⋊31D10, C23⋊D10⋊32C2, (D4×C10)⋊48C22, (C22×D20)⋊21C2, C4⋊Dic5⋊66C22, (Q8×C10)⋊41C22, C20.23D4⋊32C2, (C2×C10).316C24, (C2×C20).653C23, (C22×C20)⋊32C22, C5⋊7(C22.29C24), (C4×Dic5)⋊45C22, C10.166(C22×D4), C2.70(D4⋊8D10), D10⋊C4⋊38C22, (C2×D20).289C22, (C23×D5).80C22, C23.212(C22×D5), C22.325(C23×D5), C23.21D10⋊39C2, (C22×C10).242C23, (C2×Dic5).163C23, (C22×D5).138C23, C23.D5.136C22, (C2×C4○D4)⋊8D5, (C10×C4○D4)⋊8C2, (C2×C4)⋊7(C5⋊D4), C4.33(C2×C5⋊D4), (C2×C10).82(C2×D4), (C2×C5⋊D4)⋊31C22, C2.39(C22×C5⋊D4), C22.24(C2×C5⋊D4), (C2×C4).251(C22×D5), SmallGroup(320,1502)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1454 in 334 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×22], Q8 [×2], C23, C23 [×2], C23 [×12], D5 [×4], C10, C10 [×2], C10 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×4], C20 [×4], C20 [×2], D10 [×20], C2×C10, C2×C10 [×2], C2×C10 [×8], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, D20 [×8], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×4], C22×D5 [×8], C22×C10, C22×C10 [×2], C22.29C24, C4×Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C2×D20 [×4], C2×D20 [×4], C2×C5⋊D4 [×8], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C23×D5 [×2], C23.21D10, C20⋊7D4 [×4], C23⋊D10 [×4], C20⋊D4 [×2], C20.23D4 [×2], C22×D20, C10×C4○D4, C10.1462+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], C5⋊D4 [×4], C22×D5 [×7], C22.29C24, C2×C5⋊D4 [×6], C23×D5, D4⋊8D10 [×2], C22×C5⋊D4, C10.1462+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=a5b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a5b-1, dbd-1=a5b, be=eb, cd=dc, ece=a5c, ede=a5b2d >
►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 11 30 71)(2 12 21 72)(3 13 22 73)(4 14 23 74)(5 15 24 75)(6 16 25 76)(7 17 26 77)(8 18 27 78)(9 19 28 79)(10 20 29 80)(31 70 45 56)(32 61 46 57)(33 62 47 58)(34 63 48 59)(35 64 49 60)(36 65 50 51)(37 66 41 52)(38 67 42 53)(39 68 43 54)(40 69 44 55)
(1 71 6 76)(2 80 7 75)(3 79 8 74)(4 78 9 73)(5 77 10 72)(11 25 16 30)(12 24 17 29)(13 23 18 28)(14 22 19 27)(15 21 20 26)(31 51 36 56)(32 60 37 55)(33 59 38 54)(34 58 39 53)(35 57 40 52)(41 69 46 64)(42 68 47 63)(43 67 48 62)(44 66 49 61)(45 65 50 70)
(1 50 25 31)(2 49 26 40)(3 48 27 39)(4 47 28 38)(5 46 29 37)(6 45 30 36)(7 44 21 35)(8 43 22 34)(9 42 23 33)(10 41 24 32)(11 56 76 65)(12 55 77 64)(13 54 78 63)(14 53 79 62)(15 52 80 61)(16 51 71 70)(17 60 72 69)(18 59 73 68)(19 58 74 67)(20 57 75 66)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 46)(22 47)(23 48)(24 49)(25 50)(26 41)(27 42)(28 43)(29 44)(30 45)(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), (1,11,30,71)(2,12,21,72)(3,13,22,73)(4,14,23,74)(5,15,24,75)(6,16,25,76)(7,17,26,77)(8,18,27,78)(9,19,28,79)(10,20,29,80)(31,70,45,56)(32,61,46,57)(33,62,47,58)(34,63,48,59)(35,64,49,60)(36,65,50,51)(37,66,41,52)(38,67,42,53)(39,68,43,54)(40,69,44,55), (1,71,6,76)(2,80,7,75)(3,79,8,74)(4,78,9,73)(5,77,10,72)(11,25,16,30)(12,24,17,29)(13,23,18,28)(14,22,19,27)(15,21,20,26)(31,51,36,56)(32,60,37,55)(33,59,38,54)(34,58,39,53)(35,57,40,52)(41,69,46,64)(42,68,47,63)(43,67,48,62)(44,66,49,61)(45,65,50,70), (1,50,25,31)(2,49,26,40)(3,48,27,39)(4,47,28,38)(5,46,29,37)(6,45,30,36)(7,44,21,35)(8,43,22,34)(9,42,23,33)(10,41,24,32)(11,56,76,65)(12,55,77,64)(13,54,78,63)(14,53,79,62)(15,52,80,61)(16,51,71,70)(17,60,72,69)(18,59,73,68)(19,58,74,67)(20,57,75,66), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(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), (1,11,30,71)(2,12,21,72)(3,13,22,73)(4,14,23,74)(5,15,24,75)(6,16,25,76)(7,17,26,77)(8,18,27,78)(9,19,28,79)(10,20,29,80)(31,70,45,56)(32,61,46,57)(33,62,47,58)(34,63,48,59)(35,64,49,60)(36,65,50,51)(37,66,41,52)(38,67,42,53)(39,68,43,54)(40,69,44,55), (1,71,6,76)(2,80,7,75)(3,79,8,74)(4,78,9,73)(5,77,10,72)(11,25,16,30)(12,24,17,29)(13,23,18,28)(14,22,19,27)(15,21,20,26)(31,51,36,56)(32,60,37,55)(33,59,38,54)(34,58,39,53)(35,57,40,52)(41,69,46,64)(42,68,47,63)(43,67,48,62)(44,66,49,61)(45,65,50,70), (1,50,25,31)(2,49,26,40)(3,48,27,39)(4,47,28,38)(5,46,29,37)(6,45,30,36)(7,44,21,35)(8,43,22,34)(9,42,23,33)(10,41,24,32)(11,56,76,65)(12,55,77,64)(13,54,78,63)(14,53,79,62)(15,52,80,61)(16,51,71,70)(17,60,72,69)(18,59,73,68)(19,58,74,67)(20,57,75,66), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(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)], [(1,11,30,71),(2,12,21,72),(3,13,22,73),(4,14,23,74),(5,15,24,75),(6,16,25,76),(7,17,26,77),(8,18,27,78),(9,19,28,79),(10,20,29,80),(31,70,45,56),(32,61,46,57),(33,62,47,58),(34,63,48,59),(35,64,49,60),(36,65,50,51),(37,66,41,52),(38,67,42,53),(39,68,43,54),(40,69,44,55)], [(1,71,6,76),(2,80,7,75),(3,79,8,74),(4,78,9,73),(5,77,10,72),(11,25,16,30),(12,24,17,29),(13,23,18,28),(14,22,19,27),(15,21,20,26),(31,51,36,56),(32,60,37,55),(33,59,38,54),(34,58,39,53),(35,57,40,52),(41,69,46,64),(42,68,47,63),(43,67,48,62),(44,66,49,61),(45,65,50,70)], [(1,50,25,31),(2,49,26,40),(3,48,27,39),(4,47,28,38),(5,46,29,37),(6,45,30,36),(7,44,21,35),(8,43,22,34),(9,42,23,33),(10,41,24,32),(11,56,76,65),(12,55,77,64),(13,54,78,63),(14,53,79,62),(15,52,80,61),(16,51,71,70),(17,60,72,69),(18,59,73,68),(19,58,74,67),(20,57,75,66)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,70),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,46),(22,47),(23,48),(24,49),(25,50),(26,41),(27,42),(28,43),(29,44),(30,45),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 28 |
| 0 | 0 | 0 | 0 | 22 | 30 |
| 0 | 0 | 11 | 28 | 0 | 0 |
| 0 | 0 | 22 | 30 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 2 |
| 0 | 0 | 0 | 0 | 5 | 27 |
| 0 | 0 | 27 | 39 | 0 | 0 |
| 0 | 0 | 36 | 14 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 7 | 40 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,0,34,0,0,0,0,6,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,22,0,0,0,0,28,30,0,0,11,22,0,0,0,0,28,30,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,27,36,0,0,0,0,39,14,0,0,14,5,0,0,0,0,2,27,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,40,34,0,0,7,7,0,0,0,0,40,34,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | 2+ (1+4) | D4⋊8D10 |
| kernel | C10.1462+ (1+4) | C23.21D10 | C20⋊7D4 | C23⋊D10 | C20⋊D4 | C20.23D4 | C22×D20 | C10×C4○D4 | C2×C20 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 4 | 2 | 6 | 6 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}._{146}2_+^{(1+4)} % in TeX
G:=Group("C10.146ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1502);
// by ID
G=gap.SmallGroup(320,1502);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=a^5*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^5*b^-1,d*b*d^-1=a^5*b,b*e=e*b,c*d=d*c,e*c*e=a^5*c,e*d*e=a^5*b^2*d>;
// generators/relations