G = C10.382+ 1+4 order 320 = 26·5
38th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.382+ 1+4, (C4×D5)⋊1D4, C4⋊C4⋊22D10, (C2×D4)⋊7D10, C4⋊D4⋊10D5, C4.183(D4×D5), C23⋊D10⋊9C2, C20⋊7D4⋊33C2, C20⋊D4⋊16C2, C20⋊2D4⋊18C2, C4⋊D20⋊21C2, C22⋊C4⋊10D10, D10.75(C2×D4), C20.227(C2×D4), (C22×C4)⋊16D10, C22⋊D20⋊12C2, (D4×C10)⋊12C22, (C2×D20)⋊23C22, C4⋊Dic5⋊31C22, Dic5.18(C2×D4), C10.66(C22×D4), (C2×C10).151C24, (C2×C20).173C23, (C22×C20)⋊20C22, C5⋊3(C22.29C24), (C4×Dic5)⋊21C22, C23.D5⋊23C22, C2.40(D4⋊6D10), C2.27(D4⋊8D10), D10⋊C4⋊17C22, Dic5.5D4⋊18C2, (C2×Dic10)⋊54C22, (C22×C10).20C23, (C2×Dic5).72C23, (C23×D5).46C22, C22.172(C23×D5), C23.112(C22×D5), (C22×D5).196C23, (C2×D4×D5)⋊10C2, C2.39(C2×D4×D5), (C2×C4×D5)⋊13C22, C4⋊C4⋊7D5⋊20C2, (C2×C4○D20)⋊20C2, (C5×C4⋊D4)⋊13C2, (C5×C4⋊C4)⋊10C22, (C2×C5⋊D4)⋊14C22, (C2×C4).38(C22×D5), (C5×C22⋊C4)⋊12C22, SmallGroup(320,1279)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.382+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=a5c, ede=b2d >
Subgroups: 1534 in 334 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C42⋊C2, C22≀C2, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, Dic10, C4×D5, 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, C22×C10, C22.29C24, C4×Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, C4○D20, D4×D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C22⋊D20, Dic5.5D4, C4⋊C4⋊7D5, C4⋊D20, C20⋊7D4, C23⋊D10, C20⋊2D4, C20⋊D4, C5×C4⋊D4, C2×C4○D20, C2×D4×D5, C10.382+ 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, C22.29C24, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, D4⋊8D10, C10.382+ 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 13 53)(2 44 14 54)(3 45 15 55)(4 46 16 56)(5 47 17 57)(6 48 18 58)(7 49 19 59)(8 50 20 60)(9 41 11 51)(10 42 12 52)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 61 36 71)(27 62 37 72)(28 63 38 73)(29 64 39 74)(30 65 40 75)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 38 13 28)(2 37 14 27)(3 36 15 26)(4 35 16 25)(5 34 17 24)(6 33 18 23)(7 32 19 22)(8 31 20 21)(9 40 11 30)(10 39 12 29)(41 75 51 65)(42 74 52 64)(43 73 53 63)(44 72 54 62)(45 71 55 61)(46 80 56 70)(47 79 57 69)(48 78 58 68)(49 77 59 67)(50 76 60 66)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(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), (1,43,13,53)(2,44,14,54)(3,45,15,55)(4,46,16,56)(5,47,17,57)(6,48,18,58)(7,49,19,59)(8,50,20,60)(9,41,11,51)(10,42,12,52)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,61,36,71)(27,62,37,72)(28,63,38,73)(29,64,39,74)(30,65,40,75), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,38,13,28)(2,37,14,27)(3,36,15,26)(4,35,16,25)(5,34,17,24)(6,33,18,23)(7,32,19,22)(8,31,20,21)(9,40,11,30)(10,39,12,29)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,80,56,70)(47,79,57,69)(48,78,58,68)(49,77,59,67)(50,76,60,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(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), (1,43,13,53)(2,44,14,54)(3,45,15,55)(4,46,16,56)(5,47,17,57)(6,48,18,58)(7,49,19,59)(8,50,20,60)(9,41,11,51)(10,42,12,52)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,61,36,71)(27,62,37,72)(28,63,38,73)(29,64,39,74)(30,65,40,75), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,38,13,28)(2,37,14,27)(3,36,15,26)(4,35,16,25)(5,34,17,24)(6,33,18,23)(7,32,19,22)(8,31,20,21)(9,40,11,30)(10,39,12,29)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,80,56,70)(47,79,57,69)(48,78,58,68)(49,77,59,67)(50,76,60,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(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)], [(1,43,13,53),(2,44,14,54),(3,45,15,55),(4,46,16,56),(5,47,17,57),(6,48,18,58),(7,49,19,59),(8,50,20,60),(9,41,11,51),(10,42,12,52),(21,66,31,76),(22,67,32,77),(23,68,33,78),(24,69,34,79),(25,70,35,80),(26,61,36,71),(27,62,37,72),(28,63,38,73),(29,64,39,74),(30,65,40,75)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,38,13,28),(2,37,14,27),(3,36,15,26),(4,35,16,25),(5,34,17,24),(6,33,18,23),(7,32,19,22),(8,31,20,21),(9,40,11,30),(10,39,12,29),(41,75,51,65),(42,74,52,64),(43,73,53,63),(44,72,54,62),(45,71,55,61),(46,80,56,70),(47,79,57,69),(48,78,58,68),(49,77,59,67),(50,76,60,66)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75)]])
50 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 | 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 | 4 | 4 | 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 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | 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 | C10.382+ 1+4 | C22⋊D20 | Dic5.5D4 | C4⋊C4⋊7D5 | C4⋊D20 | C20⋊7D4 | C23⋊D10 | C20⋊2D4 | C20⋊D4 | C5×C4⋊D4 | C2×C4○D20 | C2×D4×D5 | C4×D5 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 6 | 2 | 4 | 4 | 4 |
Matrix representation of C10.382+ 1+4 ►in 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 | 35 | 35 |
| 0 | 0 | 0 | 0 | 6 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 2 | 0 |
| 0 | 0 | 6 | 23 | 0 | 2 |
| 0 | 0 | 40 | 0 | 23 | 6 |
| 0 | 0 | 0 | 40 | 35 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 6 | 40 | 0 |
| 0 | 0 | 35 | 18 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 0 | 0 |
| 0 | 0 | 20 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 | 18 | 35 |
| 0 | 0 | 6 | 40 | 20 | 23 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 0 | 0 |
| 0 | 0 | 6 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 35 |
| 0 | 0 | 0 | 0 | 6 | 23 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,6,40,0,0,0,35,23,0,40,0,0,2,0,23,35,0,0,0,2,6,18],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,23,35,0,0,0,1,6,18,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,18,20,1,6,0,0,35,23,0,40,0,0,0,0,18,20,0,0,0,0,35,23],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,0,0,0,0,18,6,0,0,0,0,35,23] >;
C10.382+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{38}2_+^{1+4} % in TeX
G:=Group("C10.38ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1279);
// by ID
G=gap.SmallGroup(320,1279);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=b^2*d>;
// generators/relations