G = C10.1202+ 1+4 order 320 = 26·5
29th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.1202+ 1+4, C4⋊C4⋊14D10, (C22×D5)⋊8D4, C4⋊D20⋊29C2, C22⋊C4⋊14D10, D10.17(C2×D4), (C22×C4)⋊22D10, C5⋊4(C23⋊3D4), C23⋊D10⋊15C2, C22⋊D20⋊18C2, D10⋊D4⋊28C2, C22.44(D4×D5), (C2×D4).162D10, (C2×D20)⋊47C22, (C22×D20)⋊10C2, (C2×C20).70C23, C10.82(C22×D4), C22.D4⋊3D5, (C2×C10).197C24, (C22×C20)⋊12C22, C10.D4⋊4C22, (C23×D5)⋊11C22, D10.13D4⋊26C2, C23.D5⋊29C22, C2.40(D4⋊8D10), D10⋊C4⋊21C22, (D4×C10).135C22, C23.23D10⋊7C2, (C22×C10).32C23, C22.218(C23×D5), C23.200(C22×D5), (C2×Dic5).101C23, (C22×D5).215C23, (C2×D4×D5)⋊14C2, C2.55(C2×D4×D5), (C2×C4×D5)⋊20C22, (C5×C4⋊C4)⋊24C22, (D5×C22⋊C4)⋊11C2, (C2×C10).58(C2×D4), (C2×C5⋊D4)⋊18C22, (C5×C22⋊C4)⋊20C22, (C2×C4).189(C22×D5), (C5×C22.D4)⋊5C2, SmallGroup(320,1325)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.1202+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=a5b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=a5b-1, dbd-1=ebe=a5b, cd=dc, ce=ec, ede=a5b2d >
Subgroups: 1694 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.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×D5, C22×C10, C23⋊3D4, C10.D4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×D20, D4×D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, D5×C22⋊C4, C22⋊D20, C22⋊D20, D10⋊D4, D10.13D4, C4⋊D20, C23.23D10, C23⋊D10, C5×C22.D4, C22×D20, C2×D4×D5, C10.1202+ 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⋊8D10, C10.1202+ 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 68 18 73)(2 69 19 74)(3 70 20 75)(4 61 11 76)(5 62 12 77)(6 63 13 78)(7 64 14 79)(8 65 15 80)(9 66 16 71)(10 67 17 72)(21 56 36 41)(22 57 37 42)(23 58 38 43)(24 59 39 44)(25 60 40 45)(26 51 31 46)(27 52 32 47)(28 53 33 48)(29 54 34 49)(30 55 35 50)
(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 13 23)(2 34 14 24)(3 35 15 25)(4 36 16 26)(5 37 17 27)(6 38 18 28)(7 39 19 29)(8 40 20 30)(9 31 11 21)(10 32 12 22)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(1 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 30)(10 29)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(41 75)(42 74)(43 73)(44 72)(45 71)(46 80)(47 79)(48 78)(49 77)(50 76)(51 65)(52 64)(53 63)(54 62)(55 61)(56 70)(57 69)(58 68)(59 67)(60 66)
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), (1,68,18,73)(2,69,19,74)(3,70,20,75)(4,61,11,76)(5,62,12,77)(6,63,13,78)(7,64,14,79)(8,65,15,80)(9,66,16,71)(10,67,17,72)(21,56,36,41)(22,57,37,42)(23,58,38,43)(24,59,39,44)(25,60,40,45)(26,51,31,46)(27,52,32,47)(28,53,33,48)(29,54,34,49)(30,55,35,50), (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), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,36,16,26)(5,37,17,27)(6,38,18,28)(7,39,19,29)(8,40,20,30)(9,31,11,21)(10,32,12,22)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,65)(52,64)(53,63)(54,62)(55,61)(56,70)(57,69)(58,68)(59,67)(60,66)>;
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), (1,68,18,73)(2,69,19,74)(3,70,20,75)(4,61,11,76)(5,62,12,77)(6,63,13,78)(7,64,14,79)(8,65,15,80)(9,66,16,71)(10,67,17,72)(21,56,36,41)(22,57,37,42)(23,58,38,43)(24,59,39,44)(25,60,40,45)(26,51,31,46)(27,52,32,47)(28,53,33,48)(29,54,34,49)(30,55,35,50), (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), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,36,16,26)(5,37,17,27)(6,38,18,28)(7,39,19,29)(8,40,20,30)(9,31,11,21)(10,32,12,22)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,65)(52,64)(53,63)(54,62)(55,61)(56,70)(57,69)(58,68)(59,67)(60,66) );
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)], [(1,68,18,73),(2,69,19,74),(3,70,20,75),(4,61,11,76),(5,62,12,77),(6,63,13,78),(7,64,14,79),(8,65,15,80),(9,66,16,71),(10,67,17,72),(21,56,36,41),(22,57,37,42),(23,58,38,43),(24,59,39,44),(25,60,40,45),(26,51,31,46),(27,52,32,47),(28,53,33,48),(29,54,34,49),(30,55,35,50)], [(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)], [(1,33,13,23),(2,34,14,24),(3,35,15,25),(4,36,16,26),(5,37,17,27),(6,38,18,28),(7,39,19,29),(8,40,20,30),(9,31,11,21),(10,32,12,22),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,71,56,61),(47,72,57,62),(48,73,58,63),(49,74,59,64),(50,75,60,65)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,30),(10,29),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(41,75),(42,74),(43,73),(44,72),(45,71),(46,80),(47,79),(48,78),(49,77),(50,76),(51,65),(52,64),(53,63),(54,62),(55,61),(56,70),(57,69),(58,68),(59,67),(60,66)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4E | 4F | 4G | 4H | 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 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 20 | 20 | 20 | 4 | ··· | 4 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D4⋊8D10 |
| kernel | C10.1202+ 1+4 | D5×C22⋊C4 | C22⋊D20 | D10⋊D4 | D10.13D4 | C4⋊D20 | C23.23D10 | C23⋊D10 | C5×C22.D4 | C22×D20 | C2×D4×D5 | C22×D5 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C2 |
| # reps | 1 | 1 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 6 | 4 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C10.1202+ 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 14 | 0 | 7 | 7 |
| 0 | 0 | 0 | 27 | 34 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 29 | 27 | 39 |
| 0 | 0 | 25 | 22 | 27 | 27 |
| 0 | 0 | 13 | 9 | 19 | 12 |
| 0 | 0 | 21 | 13 | 16 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 17 | 40 | 0 |
| 0 | 0 | 35 | 33 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 32 | 0 | 0 |
| 0 | 0 | 9 | 30 | 0 | 0 |
| 0 | 0 | 23 | 0 | 30 | 32 |
| 0 | 0 | 0 | 18 | 9 | 11 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 30 | 27 | 0 | 0 |
| 0 | 0 | 5 | 26 | 30 | 32 |
| 0 | 0 | 20 | 13 | 27 | 11 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,14,0,0,0,7,7,0,27,0,0,0,0,7,34,0,0,0,0,7,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,20,25,13,21,0,0,29,22,9,13,0,0,27,27,19,16,0,0,39,27,12,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,33,35,0,0,0,1,17,33,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,11,9,23,0,0,0,32,30,0,18,0,0,0,0,30,9,0,0,0,0,32,11],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,14,30,5,20,0,0,14,27,26,13,0,0,0,0,30,27,0,0,0,0,32,11] >;
C10.1202+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{120}2_+^{1+4} % in TeX
G:=Group("C10.120ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1325);
// by ID
G=gap.SmallGroup(320,1325);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=a^5*b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c=a^5*b^-1,d*b*d^-1=e*b*e=a^5*b,c*d=d*c,c*e=e*c,e*d*e=a^5*b^2*d>;
// generators/relations