G = C10.372+ 1+4 order 320 = 26·5
37th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.372+ 1+4, C4⋊C4⋊5D10, C4⋊D4⋊9D5, (C2×D4)⋊21D10, C22⋊C4⋊9D10, (C22×D5)⋊7D4, C20⋊7D4⋊44C2, C22.3(D4×D5), D10.73(C2×D4), (C2×D20)⋊7C22, (C22×C4)⋊15D10, C5⋊3(C23⋊3D4), C22⋊D20⋊11C2, C23⋊D10⋊22C2, (D4×C10)⋊29C22, (C2×C20).38C23, C4⋊Dic5⋊11C22, C10.64(C22×D4), (C23×D5)⋊9C22, Dic5⋊D4⋊12C2, (C2×C10).149C24, (C22×C20)⋊30C22, D10.13D4⋊11C2, D10.12D4⋊17C2, C2.39(D4⋊6D10), C2.26(D4⋊8D10), C23.D5⋊21C22, D10⋊C4⋊66C22, C10.D4⋊15C22, (C22×C10).18C23, (C2×Dic5).70C23, C22.170(C23×D5), C23.180(C22×D5), (C22×Dic5)⋊18C22, (C22×D5).194C23, (C2×D4×D5)⋊9C2, C2.37(C2×D4×D5), (C5×C4⋊C4)⋊8C22, (C2×C10).5(C2×D4), (C2×C4×D5)⋊12C22, (C5×C4⋊D4)⋊11C2, (C2×C5⋊D4)⋊13C22, (C2×D10⋊C4)⋊25C2, (C5×C22⋊C4)⋊10C22, (C2×C4).175(C22×D5), SmallGroup(320,1277)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.372+ 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, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 1630 in 346 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C22×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C23⋊3D4, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, D4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C23×D5, C22⋊D20, D10.12D4, D10.13D4, C2×D10⋊C4, C20⋊7D4, C23⋊D10, Dic5⋊D4, C5×C4⋊D4, C2×D4×D5, C10.372+ 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, C23⋊3D4, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, D4⋊8D10, C10.372+ 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 78 13 68)(2 79 14 69)(3 80 15 70)(4 71 16 61)(5 72 17 62)(6 73 18 63)(7 74 19 64)(8 75 20 65)(9 76 11 66)(10 77 12 67)(21 56 31 46)(22 57 32 47)(23 58 33 48)(24 59 34 49)(25 60 35 50)(26 51 36 41)(27 52 37 42)(28 53 38 43)(29 54 39 44)(30 55 40 45)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 41)(10 42)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 78 18 63)(2 77 19 62)(3 76 20 61)(4 75 11 70)(5 74 12 69)(6 73 13 68)(7 72 14 67)(8 71 15 66)(9 80 16 65)(10 79 17 64)(21 45 36 60)(22 44 37 59)(23 43 38 58)(24 42 39 57)(25 41 40 56)(26 50 31 55)(27 49 32 54)(28 48 33 53)(29 47 34 52)(30 46 35 51)
(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 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 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), (1,78,13,68)(2,79,14,69)(3,80,15,70)(4,71,16,61)(5,72,17,62)(6,73,18,63)(7,74,19,64)(8,75,20,65)(9,76,11,66)(10,77,12,67)(21,56,31,46)(22,57,32,47)(23,58,33,48)(24,59,34,49)(25,60,35,50)(26,51,36,41)(27,52,37,42)(28,53,38,43)(29,54,39,44)(30,55,40,45), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,41)(10,42)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,78,18,63)(2,77,19,62)(3,76,20,61)(4,75,11,70)(5,74,12,69)(6,73,13,68)(7,72,14,67)(8,71,15,66)(9,80,16,65)(10,79,17,64)(21,45,36,60)(22,44,37,59)(23,43,38,58)(24,42,39,57)(25,41,40,56)(26,50,31,55)(27,49,32,54)(28,48,33,53)(29,47,34,52)(30,46,35,51), (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)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75)>;
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), (1,78,13,68)(2,79,14,69)(3,80,15,70)(4,71,16,61)(5,72,17,62)(6,73,18,63)(7,74,19,64)(8,75,20,65)(9,76,11,66)(10,77,12,67)(21,56,31,46)(22,57,32,47)(23,58,33,48)(24,59,34,49)(25,60,35,50)(26,51,36,41)(27,52,37,42)(28,53,38,43)(29,54,39,44)(30,55,40,45), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,41)(10,42)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,78,18,63)(2,77,19,62)(3,76,20,61)(4,75,11,70)(5,74,12,69)(6,73,13,68)(7,72,14,67)(8,71,15,66)(9,80,16,65)(10,79,17,64)(21,45,36,60)(22,44,37,59)(23,43,38,58)(24,42,39,57)(25,41,40,56)(26,50,31,55)(27,49,32,54)(28,48,33,53)(29,47,34,52)(30,46,35,51), (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)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75) );
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)], [(1,78,13,68),(2,79,14,69),(3,80,15,70),(4,71,16,61),(5,72,17,62),(6,73,18,63),(7,74,19,64),(8,75,20,65),(9,76,11,66),(10,77,12,67),(21,56,31,46),(22,57,32,47),(23,58,33,48),(24,59,34,49),(25,60,35,50),(26,51,36,41),(27,52,37,42),(28,53,38,43),(29,54,39,44),(30,55,40,45)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,41),(10,42),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)], [(1,78,18,63),(2,77,19,62),(3,76,20,61),(4,75,11,70),(5,74,12,69),(6,73,13,68),(7,72,14,67),(8,71,15,66),(9,80,16,65),(10,79,17,64),(21,45,36,60),(22,44,37,59),(23,43,38,58),(24,42,39,57),(25,41,40,56),(26,50,31,55),(27,49,32,54),(28,48,33,53),(29,47,34,52),(30,46,35,51)], [(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),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 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 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D4⋊6D10 | D4⋊8D10 |
| kernel | C10.372+ 1+4 | C22⋊D20 | D10.12D4 | D10.13D4 | C2×D10⋊C4 | C20⋊7D4 | C23⋊D10 | Dic5⋊D4 | C5×C4⋊D4 | C2×D4×D5 | C22×D5 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 6 | 2 | 4 | 4 | 4 |
Matrix representation of C10.372+ 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 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 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 |
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],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,11,32,0,0,0,0,9,30,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,30,27,0,0,0,0,32,11,0,0,0,0,0,0,11,14,0,0,0,0,9,30],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
C10.372+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{37}2_+^{1+4} % in TeX
G:=Group("C10.37ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1277);
// by ID
G=gap.SmallGroup(320,1277);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,297,136,12550]);
// 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,d*b*d^-1=e*b*e^-1=a^5*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations