metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.42D10, C10.912+ (1+4), (C5×D4)⋊17D4, D4⋊8(C5⋊D4), C20⋊2D4⋊41C2, C5⋊10(D4⋊5D4), (D4×Dic5)⋊39C2, C20.253(C2×D4), (C22×D4)⋊12D5, (C2×D4).231D10, C24⋊2D5⋊14C2, C4⋊Dic5⋊45C22, Dic5⋊D4⋊42C2, C20.48D4⋊37C2, C20.17D4⋊29C2, C22⋊5(D4⋊2D5), (C2×C20).546C23, (C2×C10).301C24, (C4×Dic5)⋊43C22, (C22×C4).273D10, C10.148(C22×D4), C23.D5⋊40C22, C2.94(D4⋊6D10), D10⋊C4⋊73C22, (C2×Dic10)⋊42C22, (D4×C10).312C22, C10.D4⋊39C22, (C23×C10).80C22, C23.236(C22×D5), C22.314(C23×D5), C23.18D10⋊30C2, (C22×C20).278C22, (C22×C10).235C23, (C2×Dic5).296C23, (C22×Dic5)⋊35C22, (C22×D5).132C23, (D4×C2×C10)⋊8C2, (C4×C5⋊D4)⋊26C2, (C2×C4×D5)⋊32C22, C4.68(C2×C5⋊D4), (C2×C10).74(C2×D4), (C2×D4⋊2D5)⋊27C2, (C2×C10)⋊15(C4○D4), C22.3(C2×C5⋊D4), C10.107(C2×C4○D4), C2.71(C2×D4⋊2D5), (C2×C5⋊D4)⋊29C22, (C2×C23.D5)⋊31C2, C2.21(C22×C5⋊D4), (C2×C4).239(C22×D5), SmallGroup(320,1478)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1078 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], C5, C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×11], D5, C10 [×3], C10 [×8], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×7], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C22⋊C4 [×2], 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 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×4], C22×C10 [×10], D4⋊5D4, C4×Dic5, C10.D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4, C23.D5, C23.D5 [×10], C2×Dic10, C2×C4×D5, D4⋊2D5 [×4], C22×Dic5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×4], C22×C20, D4×C10 [×2], D4×C10 [×2], D4×C10 [×4], C23×C10 [×2], C20.48D4, C4×C5⋊D4, D4×Dic5, C23.18D10 [×2], C20.17D4, C20⋊2D4, Dic5⋊D4 [×2], C2×C23.D5 [×2], C24⋊2D5 [×2], C2×D4⋊2D5, D4×C2×C10, C24.42D10
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), C5⋊D4 [×4], C22×D5 [×7], D4⋊5D4, D4⋊2D5 [×2], C2×C5⋊D4 [×6], C23×D5, C2×D4⋊2D5, D4⋊6D10, C22×C5⋊D4, C24.42D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, 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 60)(2 51)(3 42)(4 53)(5 44)(6 55)(7 46)(8 57)(9 48)(10 59)(11 50)(12 41)(13 52)(14 43)(15 54)(16 45)(17 56)(18 47)(19 58)(20 49)(21 63)(22 74)(23 65)(24 76)(25 67)(26 78)(27 69)(28 80)(29 71)(30 62)(31 73)(32 64)(33 75)(34 66)(35 77)(36 68)(37 79)(38 70)(39 61)(40 72)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 60)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 61)(40 62)
(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 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 39 11 29)(2 28 12 38)(3 37 13 27)(4 26 14 36)(5 35 15 25)(6 24 16 34)(7 33 17 23)(8 22 18 32)(9 31 19 21)(10 40 20 30)(41 70 51 80)(42 79 52 69)(43 68 53 78)(44 77 54 67)(45 66 55 76)(46 75 56 65)(47 64 57 74)(48 73 58 63)(49 62 59 72)(50 71 60 61)
G:=sub<Sym(80)| (1,60)(2,51)(3,42)(4,53)(5,44)(6,55)(7,46)(8,57)(9,48)(10,59)(11,50)(12,41)(13,52)(14,43)(15,54)(16,45)(17,56)(18,47)(19,58)(20,49)(21,63)(22,74)(23,65)(24,76)(25,67)(26,78)(27,69)(28,80)(29,71)(30,62)(31,73)(32,64)(33,75)(34,66)(35,77)(36,68)(37,79)(38,70)(39,61)(40,72), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,60)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,61)(40,62), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(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), (1,39,11,29)(2,28,12,38)(3,37,13,27)(4,26,14,36)(5,35,15,25)(6,24,16,34)(7,33,17,23)(8,22,18,32)(9,31,19,21)(10,40,20,30)(41,70,51,80)(42,79,52,69)(43,68,53,78)(44,77,54,67)(45,66,55,76)(46,75,56,65)(47,64,57,74)(48,73,58,63)(49,62,59,72)(50,71,60,61)>;
G:=Group( (1,60)(2,51)(3,42)(4,53)(5,44)(6,55)(7,46)(8,57)(9,48)(10,59)(11,50)(12,41)(13,52)(14,43)(15,54)(16,45)(17,56)(18,47)(19,58)(20,49)(21,63)(22,74)(23,65)(24,76)(25,67)(26,78)(27,69)(28,80)(29,71)(30,62)(31,73)(32,64)(33,75)(34,66)(35,77)(36,68)(37,79)(38,70)(39,61)(40,72), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,60)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,61)(40,62), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(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), (1,39,11,29)(2,28,12,38)(3,37,13,27)(4,26,14,36)(5,35,15,25)(6,24,16,34)(7,33,17,23)(8,22,18,32)(9,31,19,21)(10,40,20,30)(41,70,51,80)(42,79,52,69)(43,68,53,78)(44,77,54,67)(45,66,55,76)(46,75,56,65)(47,64,57,74)(48,73,58,63)(49,62,59,72)(50,71,60,61) );
G=PermutationGroup([(1,60),(2,51),(3,42),(4,53),(5,44),(6,55),(7,46),(8,57),(9,48),(10,59),(11,50),(12,41),(13,52),(14,43),(15,54),(16,45),(17,56),(18,47),(19,58),(20,49),(21,63),(22,74),(23,65),(24,76),(25,67),(26,78),(27,69),(28,80),(29,71),(30,62),(31,73),(32,64),(33,75),(34,66),(35,77),(36,68),(37,79),(38,70),(39,61),(40,72)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,60),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,61),(40,62)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(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)], [(1,39,11,29),(2,28,12,38),(3,37,13,27),(4,26,14,36),(5,35,15,25),(6,24,16,34),(7,33,17,23),(8,22,18,32),(9,31,19,21),(10,40,20,30),(41,70,51,80),(42,79,52,69),(43,68,53,78),(44,77,54,67),(45,66,55,76),(46,75,56,65),(47,64,57,74),(48,73,58,63),(49,62,59,72),(50,71,60,61)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 37 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 0 | 31 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[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],[37,0,0,0,0,10,0,0,0,0,1,1,0,0,39,40],[0,4,0,0,31,0,0,0,0,0,9,0,0,0,0,9] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 20 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | 2+ (1+4) | D4⋊2D5 | D4⋊6D10 |
| kernel | C24.42D10 | C20.48D4 | C4×C5⋊D4 | D4×Dic5 | C23.18D10 | C20.17D4 | C20⋊2D4 | Dic5⋊D4 | C2×C23.D5 | C24⋊2D5 | C2×D4⋊2D5 | D4×C2×C10 | C5×D4 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C24 | D4 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 2 | 4 | 2 | 8 | 4 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2^4._{42}D_{10} % in TeX
G:=Group("C2^4.42D10"); // GroupNames label
G:=SmallGroup(320,1478);
// by ID
G=gap.SmallGroup(320,1478);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,675,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*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