metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.72C24, C40.49C23, M4(2)⋊28D10, C8○D4⋊8D5, (C2×C8)⋊23D10, (D4×D5).1C4, (Q8×D5).1C4, D4.13(C4×D5), Q8.14(C4×D5), (C2×C40)⋊26C22, C4○D4.43D10, D20.35(C2×C4), D4⋊2D5.1C4, (C8×D5)⋊12C22, C5⋊5(Q8○M4(2)), D4.Dic5⋊8C2, Q8⋊2D5.1C4, C8.56(C22×D5), C4.71(C23×D5), C8⋊D5⋊22C22, (D5×M4(2))⋊11C2, C20.74(C22×C4), C10.56(C23×C4), C5⋊2C8.33C23, (C4×D5).73C23, D20.2C4⋊13C2, D20.3C4⋊17C2, (C2×C20).514C23, Dic10.37(C2×C4), C4○D20.52C22, D10.25(C22×C4), C4.Dic5⋊27C22, (C5×M4(2))⋊28C22, Dic5.24(C22×C4), C4.39(C2×C4×D5), (C5×C8○D4)⋊9C2, C22.5(C2×C4×D5), (D5×C4○D4).3C2, C5⋊D4.6(C2×C4), (C2×C8⋊D5)⋊28C2, C2.36(D5×C22×C4), (C4×D5).11(C2×C4), (C5×D4).31(C2×C4), (C5×Q8).33(C2×C4), (C2×C5⋊2C8)⋊13C22, (C2×C4×D5).163C22, (C2×C10).12(C22×C4), (C2×Dic5).40(C2×C4), (C5×C4○D4).44C22, (C22×D5).33(C2×C4), (C2×C4).607(C22×D5), SmallGroup(320,1422)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.72C24
G = < a,b,c,d | a40=b2=c2=d2=1, bab=a29, cac=a21, ad=da, bc=cb, bd=db, dcd=a20c >
Subgroups: 734 in 258 conjugacy classes, 147 normal (24 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C5, C8, C8 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8 [×3], C2×C8 [×9], M4(2) [×3], M4(2) [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic5, Dic5 [×3], C20, C20 [×3], D10, D10 [×3], D10 [×3], C2×C10 [×3], C2×M4(2) [×6], C8○D4, C8○D4 [×7], C2×C4○D4, C5⋊2C8, C5⋊2C8 [×3], C40, C40 [×3], Dic10 [×3], C4×D5, C4×D5 [×9], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], Q8○M4(2), C8×D5 [×6], C8⋊D5, C8⋊D5 [×9], C2×C5⋊2C8 [×3], C4.Dic5 [×3], C2×C40 [×3], C5×M4(2) [×3], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C5×C4○D4, C2×C8⋊D5 [×3], D20.3C4 [×3], D5×M4(2) [×3], D20.2C4 [×3], D4.Dic5, C5×C8○D4, D5×C4○D4, C20.72C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, C4×D5 [×4], C22×D5 [×7], Q8○M4(2), C2×C4×D5 [×6], C23×D5, D5×C22×C4, C20.72C24
►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)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 49)(42 78)(43 67)(44 56)(46 74)(47 63)(48 52)(50 70)(51 59)(53 77)(54 66)(57 73)(58 62)(60 80)(61 69)(64 76)(68 72)(71 79)
(1 55)(2 76)(3 57)(4 78)(5 59)(6 80)(7 61)(8 42)(9 63)(10 44)(11 65)(12 46)(13 67)(14 48)(15 69)(16 50)(17 71)(18 52)(19 73)(20 54)(21 75)(22 56)(23 77)(24 58)(25 79)(26 60)(27 41)(28 62)(29 43)(30 64)(31 45)(32 66)(33 47)(34 68)(35 49)(36 70)(37 51)(38 72)(39 53)(40 74)
(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
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), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,49)(42,78)(43,67)(44,56)(46,74)(47,63)(48,52)(50,70)(51,59)(53,77)(54,66)(57,73)(58,62)(60,80)(61,69)(64,76)(68,72)(71,79), (1,55)(2,76)(3,57)(4,78)(5,59)(6,80)(7,61)(8,42)(9,63)(10,44)(11,65)(12,46)(13,67)(14,48)(15,69)(16,50)(17,71)(18,52)(19,73)(20,54)(21,75)(22,56)(23,77)(24,58)(25,79)(26,60)(27,41)(28,62)(29,43)(30,64)(31,45)(32,66)(33,47)(34,68)(35,49)(36,70)(37,51)(38,72)(39,53)(40,74), (41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)>;
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), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,49)(42,78)(43,67)(44,56)(46,74)(47,63)(48,52)(50,70)(51,59)(53,77)(54,66)(57,73)(58,62)(60,80)(61,69)(64,76)(68,72)(71,79), (1,55)(2,76)(3,57)(4,78)(5,59)(6,80)(7,61)(8,42)(9,63)(10,44)(11,65)(12,46)(13,67)(14,48)(15,69)(16,50)(17,71)(18,52)(19,73)(20,54)(21,75)(22,56)(23,77)(24,58)(25,79)(26,60)(27,41)(28,62)(29,43)(30,64)(31,45)(32,66)(33,47)(34,68)(35,49)(36,70)(37,51)(38,72)(39,53)(40,74), (41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80) );
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)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,49),(42,78),(43,67),(44,56),(46,74),(47,63),(48,52),(50,70),(51,59),(53,77),(54,66),(57,73),(58,62),(60,80),(61,69),(64,76),(68,72),(71,79)], [(1,55),(2,76),(3,57),(4,78),(5,59),(6,80),(7,61),(8,42),(9,63),(10,44),(11,65),(12,46),(13,67),(14,48),(15,69),(16,50),(17,71),(18,52),(19,73),(20,54),(21,75),(22,56),(23,77),(24,58),(25,79),(26,60),(27,41),(28,62),(29,43),(30,64),(31,45),(32,66),(33,47),(34,68),(35,49),(36,70),(37,51),(38,72),(39,53),(40,74)], [(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | D10 | C4×D5 | C4×D5 | Q8○M4(2) | C20.72C24 |
| kernel | C20.72C24 | C2×C8⋊D5 | D20.3C4 | D5×M4(2) | D20.2C4 | D4.Dic5 | C5×C8○D4 | D5×C4○D4 | D4×D5 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 2 | 12 | 4 | 2 | 8 |
Matrix representation of C20.72C24 ►in GL6(𝔽41)
| 6 | 35 | 0 | 0 | 0 | 0 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 39 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 40 | 2 |
| 0 | 0 | 20 | 40 | 36 | 1 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 20 | 0 | 20 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 1 | 21 | 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 | 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [6,6,0,0,0,0,35,1,0,0,0,0,0,0,0,25,40,20,0,0,39,0,0,40,0,0,0,0,40,36,0,0,0,0,2,1],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,1,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,20,1,21,0,0,0,0,0,1,0,0,1,20,0,21,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
C20.72C24 in GAP, Magma, Sage, TeX
C_{20}._{72}C_2^4 % in TeX
G:=Group("C20.72C2^4"); // GroupNames label
G:=SmallGroup(320,1422);
// by ID
G=gap.SmallGroup(320,1422);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,80,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^29,c*a*c=a^21,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^20*c>;
// generators/relations