metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.27D10, C10.32+ (1+4), C5⋊D4⋊8D4, C5⋊1(D4⋊5D4), (C22×C4)⋊8D10, C22⋊D20⋊2C2, D10⋊D4⋊1C2, C20⋊7D4⋊17C2, C22⋊C4⋊40D10, D10.35(C2×D4), (C2×D20)⋊2C22, C24⋊2D5⋊2C2, C4⋊Dic5⋊4C22, C22.18(D4×D5), C22⋊4(C4○D20), (C2×C10).34C24, Dic5.38(C2×D4), C10.37(C22×D4), Dic5⋊4D4⋊40C2, D10.12D4⋊1C2, C23.D5⋊8C22, C2.7(D4⋊6D10), (C2×C20).128C23, (C22×C20)⋊14C22, Dic5.5D4⋊1C2, (C4×Dic5)⋊47C22, D10⋊C4⋊46C22, C22.73(C23×D5), Dic5.14D4⋊2C2, (C2×Dic10)⋊48C22, C10.D4⋊49C22, C23.23D10⋊9C2, (C23×C10).60C22, (C23×D5).31C22, C23.221(C22×D5), (C22×C10).387C23, (C2×Dic5).190C23, (C22×D5).162C23, (C22×Dic5).78C22, C2.11(C2×D4×D5), (C4×C5⋊D4)⋊1C2, (C2×C4○D20)⋊3C2, (C2×C4×D5)⋊40C22, (C2×C10)⋊8(C4○D4), (D5×C22⋊C4)⋊24C2, (C2×C22⋊C4)⋊13D5, C2.16(C2×C4○D20), C10.14(C2×C4○D4), (C22×C5⋊D4)⋊5C2, (C2×C5⋊D4)⋊1C22, (C10×C22⋊C4)⋊18C2, (C2×C10).383(C2×D4), (C5×C22⋊C4)⋊53C22, (C2×C4).259(C22×D5), SmallGroup(320,1162)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1334 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×4], C22 [×25], C5, C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], D5 [×4], C10 [×3], C10 [×5], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24, C24, Dic5 [×2], Dic5 [×4], C20 [×4], D10 [×2], D10 [×12], C2×C10, C2×C10 [×4], C2×C10 [×11], C2×C22⋊C4, C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×5], D20 [×3], C2×Dic5 [×5], C2×Dic5 [×2], C5⋊D4 [×4], C5⋊D4 [×11], C2×C20 [×4], C2×C20 [×3], C22×D5 [×3], C22×D5 [×5], C22×C10 [×3], C22×C10 [×5], D4⋊5D4, C4×Dic5, C10.D4 [×3], C4⋊Dic5, D10⋊C4 [×5], C23.D5 [×3], C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×3], C2×D20 [×2], C4○D20 [×4], C22×Dic5, C2×C5⋊D4 [×7], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, Dic5.14D4, D5×C22⋊C4, Dic5⋊4D4, C22⋊D20, D10.12D4, D10⋊D4 [×2], Dic5.5D4, C4×C5⋊D4, C23.23D10, C20⋊7D4, C24⋊2D5, C10×C22⋊C4, C2×C4○D20, C22×C5⋊D4, C24.27D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D5 [×7], D4⋊5D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4⋊6D10, C24.27D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
►On 80 points
(1 11)(2 60)(3 13)(4 42)(5 15)(6 44)(7 17)(8 46)(9 19)(10 48)(12 50)(14 52)(16 54)(18 56)(20 58)(21 31)(22 79)(23 33)(24 61)(25 35)(26 63)(27 37)(28 65)(29 39)(30 67)(32 69)(34 71)(36 73)(38 75)(40 77)(41 51)(43 53)(45 55)(47 57)(49 59)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 24 31 34)(22 33 32 23)(25 40 35 30)(26 29 36 39)(27 38 37 28)(41 46 51 56)(42 55 52 45)(43 44 53 54)(47 60 57 50)(48 49 58 59)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)
G:=sub<Sym(80)| (1,11)(2,60)(3,13)(4,42)(5,15)(6,44)(7,17)(8,46)(9,19)(10,48)(12,50)(14,52)(16,54)(18,56)(20,58)(21,31)(22,79)(23,33)(24,61)(25,35)(26,63)(27,37)(28,65)(29,39)(30,67)(32,69)(34,71)(36,73)(38,75)(40,77)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,46,51,56)(42,55,52,45)(43,44,53,54)(47,60,57,50)(48,49,58,59)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)>;
G:=Group( (1,11)(2,60)(3,13)(4,42)(5,15)(6,44)(7,17)(8,46)(9,19)(10,48)(12,50)(14,52)(16,54)(18,56)(20,58)(21,31)(22,79)(23,33)(24,61)(25,35)(26,63)(27,37)(28,65)(29,39)(30,67)(32,69)(34,71)(36,73)(38,75)(40,77)(41,51)(43,53)(45,55)(47,57)(49,59)(62,72)(64,74)(66,76)(68,78)(70,80), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,46,51,56)(42,55,52,45)(43,44,53,54)(47,60,57,50)(48,49,58,59)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70) );
G=PermutationGroup([(1,11),(2,60),(3,13),(4,42),(5,15),(6,44),(7,17),(8,46),(9,19),(10,48),(12,50),(14,52),(16,54),(18,56),(20,58),(21,31),(22,79),(23,33),(24,61),(25,35),(26,63),(27,37),(28,65),(29,39),(30,67),(32,69),(34,71),(36,73),(38,75),(40,77),(41,51),(43,53),(45,55),(47,57),(49,59),(62,72),(64,74),(66,76),(68,78),(70,80)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,24,31,34),(22,33,32,23),(25,40,35,30),(26,29,36,39),(27,38,37,28),(41,46,51,56),(42,55,52,45),(43,44,53,54),(47,60,57,50),(48,49,58,59),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 23 | 6 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 13 | 13 | 0 | 0 |
| 28 | 9 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 40 |
| 13 | 13 | 0 | 0 |
| 9 | 28 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,1,0,0,0,40],[23,35,0,0,6,18,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[13,28,0,0,13,9,0,0,0,0,1,0,0,0,39,40],[13,9,0,0,13,28,0,0,0,0,1,0,0,0,39,40] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
65 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4×D5 | D4⋊6D10 |
| kernel | C24.27D10 | Dic5.14D4 | D5×C22⋊C4 | Dic5⋊4D4 | C22⋊D20 | D10.12D4 | D10⋊D4 | Dic5.5D4 | C4×C5⋊D4 | C23.23D10 | C20⋊7D4 | C24⋊2D5 | C10×C22⋊C4 | C2×C4○D20 | C22×C5⋊D4 | C5⋊D4 | C2×C22⋊C4 | C2×C10 | C22⋊C4 | C22×C4 | C24 | C22 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 8 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2^4._{27}D_{10} % in TeX
G:=Group("C2^4.27D10"); // GroupNames label
G:=SmallGroup(320,1162);
// by ID
G=gap.SmallGroup(320,1162);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations