metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.38D10, C10.872+ (1+4), (D4×C10)⋊24C4, D4⋊6(C2×Dic5), (D4×Dic5)⋊37C2, (C2×D4)⋊11Dic5, C23⋊3(C2×Dic5), (C2×D4).251D10, C10.66(C23×C4), C4⋊Dic5⋊76C22, (C22×D4).12D5, C2.5(D4⋊6D10), C2.7(C23×Dic5), (C2×C10).293C24, C20.153(C22×C4), (C2×C20).541C23, C5⋊5(C22.11C24), (C4×Dic5)⋊40C22, (C22×C4).270D10, C23.D5⋊59C22, C4.17(C22×Dic5), C22.45(C23×D5), (D4×C10).270C22, (C23×C10).75C22, C23.204(C22×D5), C23.21D10⋊32C2, C22.1(C22×Dic5), (C22×C10).229C23, (C22×C20).274C22, (C2×Dic5).293C23, (C22×Dic5)⋊31C22, (D4×C2×C10).9C2, (C2×C20)⋊28(C2×C4), (C5×D4)⋊30(C2×C4), (C2×C4)⋊4(C2×Dic5), (C22×C10)⋊19(C2×C4), (C2×C23.D5)⋊26C2, (C2×C4).624(C22×D5), (C2×C10).128(C22×C4), SmallGroup(320,1470)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 910 in 338 conjugacy classes, 191 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C5, C2×C4 [×6], C2×C4 [×16], D4 [×16], C23, C23 [×12], C23 [×4], C10, C10 [×2], C10 [×10], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×18], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×4], C22.11C24, C4×Dic5 [×4], C4⋊Dic5 [×4], C23.D5 [×12], C22×Dic5 [×8], C22×C20, D4×C10 [×12], C23×C10 [×2], C23.21D10 [×2], D4×Dic5 [×8], C2×C23.D5 [×4], D4×C2×C10, C24.38D10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, 2+ (1+4) [×2], C2×Dic5 [×28], C22×D5 [×7], C22.11C24, C22×Dic5 [×14], C23×D5, D4⋊6D10 [×2], C23×Dic5, C24.38D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
►On 80 points
Generators in S80
(1 59)(2 55)(3 51)(4 57)(5 53)(6 69)(7 65)(8 61)(9 67)(10 63)(11 45)(12 41)(13 47)(14 43)(15 49)(16 46)(17 42)(18 48)(19 44)(20 50)(21 74)(22 80)(23 76)(24 72)(25 78)(26 79)(27 75)(28 71)(29 77)(30 73)(31 58)(32 54)(33 60)(34 56)(35 52)(36 66)(37 62)(38 68)(39 64)(40 70)
(1 28)(2 29)(3 30)(4 26)(5 27)(6 17)(7 18)(8 19)(9 20)(10 16)(11 37)(12 38)(13 39)(14 40)(15 36)(21 35)(22 31)(23 32)(24 33)(25 34)(41 68)(42 69)(43 70)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 71)(60 72)
(1 23)(2 24)(3 25)(4 21)(5 22)(6 17)(7 18)(8 19)(9 20)(10 16)(11 37)(12 38)(13 39)(14 40)(15 36)(26 35)(27 31)(28 32)(29 33)(30 34)(41 68)(42 69)(43 70)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 78)(52 79)(53 80)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)
(1 32)(2 33)(3 34)(4 35)(5 31)(6 39)(7 40)(8 36)(9 37)(10 38)(11 20)(12 16)(13 17)(14 18)(15 19)(21 26)(22 27)(23 28)(24 29)(25 30)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 36 23 15)(2 40 24 14)(3 39 25 13)(4 38 21 12)(5 37 22 11)(6 30 17 34)(7 29 18 33)(8 28 19 32)(9 27 20 31)(10 26 16 35)(41 52 68 79)(42 51 69 78)(43 60 70 77)(44 59 61 76)(45 58 62 75)(46 57 63 74)(47 56 64 73)(48 55 65 72)(49 54 66 71)(50 53 67 80)
G:=sub<Sym(80)| (1,59)(2,55)(3,51)(4,57)(5,53)(6,69)(7,65)(8,61)(9,67)(10,63)(11,45)(12,41)(13,47)(14,43)(15,49)(16,46)(17,42)(18,48)(19,44)(20,50)(21,74)(22,80)(23,76)(24,72)(25,78)(26,79)(27,75)(28,71)(29,77)(30,73)(31,58)(32,54)(33,60)(34,56)(35,52)(36,66)(37,62)(38,68)(39,64)(40,70), (1,28)(2,29)(3,30)(4,26)(5,27)(6,17)(7,18)(8,19)(9,20)(10,16)(11,37)(12,38)(13,39)(14,40)(15,36)(21,35)(22,31)(23,32)(24,33)(25,34)(41,68)(42,69)(43,70)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,71)(60,72), (1,23)(2,24)(3,25)(4,21)(5,22)(6,17)(7,18)(8,19)(9,20)(10,16)(11,37)(12,38)(13,39)(14,40)(15,36)(26,35)(27,31)(28,32)(29,33)(30,34)(41,68)(42,69)(43,70)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,78)(52,79)(53,80)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77), (1,32)(2,33)(3,34)(4,35)(5,31)(6,39)(7,40)(8,36)(9,37)(10,38)(11,20)(12,16)(13,17)(14,18)(15,19)(21,26)(22,27)(23,28)(24,29)(25,30)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,36,23,15)(2,40,24,14)(3,39,25,13)(4,38,21,12)(5,37,22,11)(6,30,17,34)(7,29,18,33)(8,28,19,32)(9,27,20,31)(10,26,16,35)(41,52,68,79)(42,51,69,78)(43,60,70,77)(44,59,61,76)(45,58,62,75)(46,57,63,74)(47,56,64,73)(48,55,65,72)(49,54,66,71)(50,53,67,80)>;
G:=Group( (1,59)(2,55)(3,51)(4,57)(5,53)(6,69)(7,65)(8,61)(9,67)(10,63)(11,45)(12,41)(13,47)(14,43)(15,49)(16,46)(17,42)(18,48)(19,44)(20,50)(21,74)(22,80)(23,76)(24,72)(25,78)(26,79)(27,75)(28,71)(29,77)(30,73)(31,58)(32,54)(33,60)(34,56)(35,52)(36,66)(37,62)(38,68)(39,64)(40,70), (1,28)(2,29)(3,30)(4,26)(5,27)(6,17)(7,18)(8,19)(9,20)(10,16)(11,37)(12,38)(13,39)(14,40)(15,36)(21,35)(22,31)(23,32)(24,33)(25,34)(41,68)(42,69)(43,70)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,71)(60,72), (1,23)(2,24)(3,25)(4,21)(5,22)(6,17)(7,18)(8,19)(9,20)(10,16)(11,37)(12,38)(13,39)(14,40)(15,36)(26,35)(27,31)(28,32)(29,33)(30,34)(41,68)(42,69)(43,70)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,78)(52,79)(53,80)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77), (1,32)(2,33)(3,34)(4,35)(5,31)(6,39)(7,40)(8,36)(9,37)(10,38)(11,20)(12,16)(13,17)(14,18)(15,19)(21,26)(22,27)(23,28)(24,29)(25,30)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,36,23,15)(2,40,24,14)(3,39,25,13)(4,38,21,12)(5,37,22,11)(6,30,17,34)(7,29,18,33)(8,28,19,32)(9,27,20,31)(10,26,16,35)(41,52,68,79)(42,51,69,78)(43,60,70,77)(44,59,61,76)(45,58,62,75)(46,57,63,74)(47,56,64,73)(48,55,65,72)(49,54,66,71)(50,53,67,80) );
G=PermutationGroup([(1,59),(2,55),(3,51),(4,57),(5,53),(6,69),(7,65),(8,61),(9,67),(10,63),(11,45),(12,41),(13,47),(14,43),(15,49),(16,46),(17,42),(18,48),(19,44),(20,50),(21,74),(22,80),(23,76),(24,72),(25,78),(26,79),(27,75),(28,71),(29,77),(30,73),(31,58),(32,54),(33,60),(34,56),(35,52),(36,66),(37,62),(38,68),(39,64),(40,70)], [(1,28),(2,29),(3,30),(4,26),(5,27),(6,17),(7,18),(8,19),(9,20),(10,16),(11,37),(12,38),(13,39),(14,40),(15,36),(21,35),(22,31),(23,32),(24,33),(25,34),(41,68),(42,69),(43,70),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,71),(60,72)], [(1,23),(2,24),(3,25),(4,21),(5,22),(6,17),(7,18),(8,19),(9,20),(10,16),(11,37),(12,38),(13,39),(14,40),(15,36),(26,35),(27,31),(28,32),(29,33),(30,34),(41,68),(42,69),(43,70),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,78),(52,79),(53,80),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77)], [(1,32),(2,33),(3,34),(4,35),(5,31),(6,39),(7,40),(8,36),(9,37),(10,38),(11,20),(12,16),(13,17),(14,18),(15,19),(21,26),(22,27),(23,28),(24,29),(25,30),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(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)], [(1,36,23,15),(2,40,24,14),(3,39,25,13),(4,38,21,12),(5,37,22,11),(6,30,17,34),(7,29,18,33),(8,28,19,32),(9,27,20,31),(10,26,16,35),(41,52,68,79),(42,51,69,78),(43,60,70,77),(44,59,61,76),(45,58,62,75),(46,57,63,74),(47,56,64,73),(48,55,65,72),(49,54,66,71),(50,53,67,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 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 | 17 | 14 | 40 | 0 |
| 0 | 0 | 14 | 17 | 0 | 40 |
| 40 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 27 | 1 | 0 |
| 0 | 0 | 14 | 0 | 0 | 40 |
| 0 | 21 | 0 | 0 | 0 | 0 |
| 39 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 23 | 26 | 0 |
| 0 | 0 | 18 | 16 | 0 | 15 |
| 0 | 0 | 20 | 0 | 16 | 23 |
| 0 | 0 | 0 | 21 | 18 | 25 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,17,14,0,0,0,1,14,17,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,14,0,0,0,40,27,0,0,0,0,0,1,0,0,0,0,0,0,40],[0,39,0,0,0,0,21,0,0,0,0,0,0,0,25,18,20,0,0,0,23,16,0,21,0,0,26,0,16,18,0,0,0,15,23,25] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4T | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | Dic5 | D10 | D10 | 2+ (1+4) | D4⋊6D10 |
| kernel | C24.38D10 | C23.21D10 | D4×Dic5 | C2×C23.D5 | D4×C2×C10 | D4×C10 | C22×D4 | C22×C4 | C2×D4 | C2×D4 | C24 | C10 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 16 | 2 | 2 | 16 | 8 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2^4._{38}D_{10} % in TeX
G:=Group("C2^4.38D10"); // GroupNames label
G:=SmallGroup(320,1470);
// by ID
G=gap.SmallGroup(320,1470);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=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^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀