?
G = C10.1222+ (1+4) order 320 = 26·5
31st non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.1222+ (1+4), C4⋊C4⋊30D10, C22⋊C4⋊17D10, (D4×Dic5)⋊28C2, D10⋊2Q8⋊32C2, C23⋊D10.2C2, (C2×D4).166D10, (C2×C20).74C23, C4⋊Dic5⋊39C22, C22.D4⋊7D5, Dic5⋊4D4⋊21C2, D10.35(C4○D4), (C2×C10).202C24, (C4×Dic5)⋊55C22, (C22×C4).258D10, D10.12D4⋊31C2, C23.D5⋊31C22, C2.43(D4⋊8D10), Dic5.5D4⋊33C2, C5⋊6(C22.45C24), (C2×Dic10)⋊30C22, (D4×C10).140C22, C22.D20⋊21C2, C10.D4⋊24C22, (C23×D5).59C22, (C22×D5).86C23, C22.223(C23×D5), C23.202(C22×D5), Dic5.14D4⋊32C2, D10⋊C4.33C22, C22.19(D4⋊2D5), C23.21D10⋊12C2, (C22×C10).222C23, (C22×C20).114C22, (C2×Dic5).255C23, (C22×Dic5)⋊26C22, C2.64(D5×C4○D4), C4⋊C4⋊D5⋊28C2, C4⋊C4⋊7D5⋊32C2, (C5×C4⋊C4)⋊28C22, (D5×C22⋊C4)⋊14C2, C10.176(C2×C4○D4), C2.53(C2×D4⋊2D5), (C2×D10⋊C4)⋊24C2, (C2×C4×D5).120C22, (C2×C10).47(C4○D4), (C5×C22⋊C4)⋊24C22, (C2×C4).297(C22×D5), (C2×C5⋊D4).47C22, (C5×C22.D4)⋊10C2, SmallGroup(320,1330)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 950 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C5, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D5 [×3], C10 [×3], C10 [×3], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×6], C20 [×5], D10 [×2], D10 [×9], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C42⋊2C2 [×2], Dic10, C4×D5 [×3], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×3], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×D5 [×5], C22×C10 [×2], C22.45C24, C4×Dic5 [×3], C10.D4 [×2], C4⋊Dic5 [×4], D10⋊C4 [×8], C23.D5 [×3], C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×2], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20, D4×C10, C23×D5, Dic5.14D4, D5×C22⋊C4, Dic5⋊4D4, D10.12D4, Dic5.5D4, C22.D20, C4⋊C4⋊7D5, D10⋊2Q8, C4⋊C4⋊D5 [×2], C23.21D10, C2×D10⋊C4, D4×Dic5, C23⋊D10, C5×C22.D4, C10.1222+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.45C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D5×C4○D4, D4⋊8D10, C10.1222+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, bab-1=eae-1=a-1, ac=ca, ad=da, cbc=b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=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 73 29 16)(2 72 30 15)(3 71 21 14)(4 80 22 13)(5 79 23 12)(6 78 24 11)(7 77 25 20)(8 76 26 19)(9 75 27 18)(10 74 28 17)(31 62 48 55)(32 61 49 54)(33 70 50 53)(34 69 41 52)(35 68 42 51)(36 67 43 60)(37 66 44 59)(38 65 45 58)(39 64 46 57)(40 63 47 56)
(1 78)(2 79)(3 80)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 29)(12 30)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 61)(40 62)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)
(1 38 29 45)(2 39 30 46)(3 40 21 47)(4 31 22 48)(5 32 23 49)(6 33 24 50)(7 34 25 41)(8 35 26 42)(9 36 27 43)(10 37 28 44)(11 58 78 65)(12 59 79 66)(13 60 80 67)(14 51 71 68)(15 52 72 69)(16 53 73 70)(17 54 74 61)(18 55 75 62)(19 56 76 63)(20 57 77 64)
(1 50 6 45)(2 49 7 44)(3 48 8 43)(4 47 9 42)(5 46 10 41)(11 65 16 70)(12 64 17 69)(13 63 18 68)(14 62 19 67)(15 61 20 66)(21 31 26 36)(22 40 27 35)(23 39 28 34)(24 38 29 33)(25 37 30 32)(51 80 56 75)(52 79 57 74)(53 78 58 73)(54 77 59 72)(55 76 60 71)
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), (1,73,29,16)(2,72,30,15)(3,71,21,14)(4,80,22,13)(5,79,23,12)(6,78,24,11)(7,77,25,20)(8,76,26,19)(9,75,27,18)(10,74,28,17)(31,62,48,55)(32,61,49,54)(33,70,50,53)(34,69,41,52)(35,68,42,51)(36,67,43,60)(37,66,44,59)(38,65,45,58)(39,64,46,57)(40,63,47,56), (1,78)(2,79)(3,80)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,29)(12,30)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,61)(40,62)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58), (1,38,29,45)(2,39,30,46)(3,40,21,47)(4,31,22,48)(5,32,23,49)(6,33,24,50)(7,34,25,41)(8,35,26,42)(9,36,27,43)(10,37,28,44)(11,58,78,65)(12,59,79,66)(13,60,80,67)(14,51,71,68)(15,52,72,69)(16,53,73,70)(17,54,74,61)(18,55,75,62)(19,56,76,63)(20,57,77,64), (1,50,6,45)(2,49,7,44)(3,48,8,43)(4,47,9,42)(5,46,10,41)(11,65,16,70)(12,64,17,69)(13,63,18,68)(14,62,19,67)(15,61,20,66)(21,31,26,36)(22,40,27,35)(23,39,28,34)(24,38,29,33)(25,37,30,32)(51,80,56,75)(52,79,57,74)(53,78,58,73)(54,77,59,72)(55,76,60,71)>;
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), (1,73,29,16)(2,72,30,15)(3,71,21,14)(4,80,22,13)(5,79,23,12)(6,78,24,11)(7,77,25,20)(8,76,26,19)(9,75,27,18)(10,74,28,17)(31,62,48,55)(32,61,49,54)(33,70,50,53)(34,69,41,52)(35,68,42,51)(36,67,43,60)(37,66,44,59)(38,65,45,58)(39,64,46,57)(40,63,47,56), (1,78)(2,79)(3,80)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,29)(12,30)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,61)(40,62)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58), (1,38,29,45)(2,39,30,46)(3,40,21,47)(4,31,22,48)(5,32,23,49)(6,33,24,50)(7,34,25,41)(8,35,26,42)(9,36,27,43)(10,37,28,44)(11,58,78,65)(12,59,79,66)(13,60,80,67)(14,51,71,68)(15,52,72,69)(16,53,73,70)(17,54,74,61)(18,55,75,62)(19,56,76,63)(20,57,77,64), (1,50,6,45)(2,49,7,44)(3,48,8,43)(4,47,9,42)(5,46,10,41)(11,65,16,70)(12,64,17,69)(13,63,18,68)(14,62,19,67)(15,61,20,66)(21,31,26,36)(22,40,27,35)(23,39,28,34)(24,38,29,33)(25,37,30,32)(51,80,56,75)(52,79,57,74)(53,78,58,73)(54,77,59,72)(55,76,60,71) );
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)], [(1,73,29,16),(2,72,30,15),(3,71,21,14),(4,80,22,13),(5,79,23,12),(6,78,24,11),(7,77,25,20),(8,76,26,19),(9,75,27,18),(10,74,28,17),(31,62,48,55),(32,61,49,54),(33,70,50,53),(34,69,41,52),(35,68,42,51),(36,67,43,60),(37,66,44,59),(38,65,45,58),(39,64,46,57),(40,63,47,56)], [(1,78),(2,79),(3,80),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,29),(12,30),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,61),(40,62),(41,59),(42,60),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(49,57),(50,58)], [(1,38,29,45),(2,39,30,46),(3,40,21,47),(4,31,22,48),(5,32,23,49),(6,33,24,50),(7,34,25,41),(8,35,26,42),(9,36,27,43),(10,37,28,44),(11,58,78,65),(12,59,79,66),(13,60,80,67),(14,51,71,68),(15,52,72,69),(16,53,73,70),(17,54,74,61),(18,55,75,62),(19,56,76,63),(20,57,77,64)], [(1,50,6,45),(2,49,7,44),(3,48,8,43),(4,47,9,42),(5,46,10,41),(11,65,16,70),(12,64,17,69),(13,63,18,68),(14,62,19,67),(15,61,20,66),(21,31,26,36),(22,40,27,35),(23,39,28,34),(24,38,29,33),(25,37,30,32),(51,80,56,75),(52,79,57,74),(53,78,58,73),(54,77,59,72),(55,76,60,71)])
Matrix representation ►G ⊆ 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,0,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | C10.1222+ (1+4) | Dic5.14D4 | D5×C22⋊C4 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C22.D20 | C4⋊C4⋊7D5 | D10⋊2Q8 | C4⋊C4⋊D5 | C23.21D10 | C2×D10⋊C4 | D4×Dic5 | C23⋊D10 | C5×C22.D4 | C22.D4 | D10 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 6 | 4 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{122}2_+^{(1+4)} % in TeX
G:=Group("C10.122ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1330);
// by ID
G=gap.SmallGroup(320,1330);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,346,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5,b*a*b^-1=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations