metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊28D10, D10⋊5(C4○D4), C22⋊C4⋊31D10, (C2×Dic5)⋊21D4, D20⋊8C4⋊32C2, D10⋊D4⋊31C2, C22⋊D20⋊20C2, C23⋊D10⋊16C2, C22.45(D4×D5), D10⋊Q8⋊29C2, (C2×D4).165D10, (C2×C20).73C23, Dic5.89(C2×D4), C10.85(C22×D4), Dic5⋊4D4⋊20C2, (C2×C10).200C24, C5⋊6(C22.19C24), (C4×Dic5)⋊32C22, (C22×C4).322D10, C22.D4⋊18D5, D10⋊C4⋊27C22, C23.27(C22×D5), (C2×Dic10)⋊28C22, (C2×D20).162C22, (D4×C10).138C22, C10.D4⋊23C22, (C22×C10).35C23, C22.221(C23×D5), C23.D5.43C22, C23.11D10⋊13C2, C23.23D10⋊21C2, (C22×C20).368C22, (C2×Dic5).104C23, (C22×Dic5)⋊25C22, (C22×D5).218C23, (C23×D5).122C22, C2.58(C2×D4×D5), C2.62(D5×C4○D4), (D5×C22×C4)⋊24C2, (C2×C4×D5)⋊22C22, (C5×C4⋊C4)⋊26C22, (C2×C10).61(C2×D4), (C2×D4⋊2D5)⋊17C2, C10.174(C2×C4○D4), (C2×C5⋊D4)⋊19C22, (C2×C4).63(C22×D5), (C5×C22⋊C4)⋊22C22, (C5×C22.D4)⋊8C2, SmallGroup(320,1328)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊28D10
G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=dad=a-1, cac-1=ab2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
Subgroups: 1262 in 330 conjugacy classes, 107 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C23×C4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22.19C24, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×C4×D5, C2×D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C23.11D10, Dic5⋊4D4, C22⋊D20, D10⋊D4, D20⋊8C4, D10⋊Q8, C23.23D10, C23⋊D10, C5×C22.D4, D5×C22×C4, C2×D4⋊2D5, C4⋊C4⋊28D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C22×D5, C22.19C24, D4×D5, C23×D5, C2×D4×D5, D5×C4○D4, C4⋊C4⋊28D10
►On 80 points
(1 70 18 65)(2 79 19 74)(3 62 20 67)(4 71 11 76)(5 64 12 69)(6 73 13 78)(7 66 14 61)(8 75 15 80)(9 68 16 63)(10 77 17 72)(21 54 26 49)(22 45 27 60)(23 56 28 41)(24 47 29 52)(25 58 30 43)(31 46 36 51)(32 57 37 42)(33 48 38 53)(34 59 39 44)(35 50 40 55)
(1 30 13 38)(2 39 14 21)(3 22 15 40)(4 31 16 23)(5 24 17 32)(6 33 18 25)(7 26 19 34)(8 35 20 27)(9 28 11 36)(10 37 12 29)(41 71 51 63)(42 64 52 72)(43 73 53 65)(44 66 54 74)(45 75 55 67)(46 68 56 76)(47 77 57 69)(48 70 58 78)(49 79 59 61)(50 62 60 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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 20)(9 19)(10 18)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 40)(28 39)(29 38)(30 37)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(50 60)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)
G:=sub<Sym(80)| (1,70,18,65)(2,79,19,74)(3,62,20,67)(4,71,11,76)(5,64,12,69)(6,73,13,78)(7,66,14,61)(8,75,15,80)(9,68,16,63)(10,77,17,72)(21,54,26,49)(22,45,27,60)(23,56,28,41)(24,47,29,52)(25,58,30,43)(31,46,36,51)(32,57,37,42)(33,48,38,53)(34,59,39,44)(35,50,40,55), (1,30,13,38)(2,39,14,21)(3,22,15,40)(4,31,16,23)(5,24,17,32)(6,33,18,25)(7,26,19,34)(8,35,20,27)(9,28,11,36)(10,37,12,29)(41,71,51,63)(42,64,52,72)(43,73,53,65)(44,66,54,74)(45,75,55,67)(46,68,56,76)(47,77,57,69)(48,70,58,78)(49,79,59,61)(50,62,60,80), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,20)(9,19)(10,18)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,40)(28,39)(29,38)(30,37)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)>;
G:=Group( (1,70,18,65)(2,79,19,74)(3,62,20,67)(4,71,11,76)(5,64,12,69)(6,73,13,78)(7,66,14,61)(8,75,15,80)(9,68,16,63)(10,77,17,72)(21,54,26,49)(22,45,27,60)(23,56,28,41)(24,47,29,52)(25,58,30,43)(31,46,36,51)(32,57,37,42)(33,48,38,53)(34,59,39,44)(35,50,40,55), (1,30,13,38)(2,39,14,21)(3,22,15,40)(4,31,16,23)(5,24,17,32)(6,33,18,25)(7,26,19,34)(8,35,20,27)(9,28,11,36)(10,37,12,29)(41,71,51,63)(42,64,52,72)(43,73,53,65)(44,66,54,74)(45,75,55,67)(46,68,56,76)(47,77,57,69)(48,70,58,78)(49,79,59,61)(50,62,60,80), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,20)(9,19)(10,18)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,40)(28,39)(29,38)(30,37)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77) );
G=PermutationGroup([[(1,70,18,65),(2,79,19,74),(3,62,20,67),(4,71,11,76),(5,64,12,69),(6,73,13,78),(7,66,14,61),(8,75,15,80),(9,68,16,63),(10,77,17,72),(21,54,26,49),(22,45,27,60),(23,56,28,41),(24,47,29,52),(25,58,30,43),(31,46,36,51),(32,57,37,42),(33,48,38,53),(34,59,39,44),(35,50,40,55)], [(1,30,13,38),(2,39,14,21),(3,22,15,40),(4,31,16,23),(5,24,17,32),(6,33,18,25),(7,26,19,34),(8,35,20,27),(9,28,11,36),(10,37,12,29),(41,71,51,63),(42,64,52,72),(43,73,53,65),(44,66,54,74),(45,75,55,67),(46,68,56,76),(47,77,57,69),(48,70,58,78),(49,79,59,61),(50,62,60,80)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,20),(9,19),(10,18),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,40),(28,39),(29,38),(30,37),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(50,60),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,80),(68,79),(69,78),(70,77)]])
56 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 | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D4×D5 | D5×C4○D4 |
| kernel | C4⋊C4⋊28D10 | C23.11D10 | Dic5⋊4D4 | C22⋊D20 | D10⋊D4 | D20⋊8C4 | D10⋊Q8 | C23.23D10 | C23⋊D10 | C5×C22.D4 | D5×C22×C4 | C2×D4⋊2D5 | C2×Dic5 | C22.D4 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 8 | 6 | 4 | 2 | 2 | 4 | 8 |
Matrix representation of C4⋊C4⋊28D10 ►in GL6(𝔽41)
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 37 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 5 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 25 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,37,32],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,16,0,0,0,0,5,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,1,25,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,16,0,0,0,0,0,1] >;
C4⋊C4⋊28D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{28}D_{10} % in TeX
G:=Group("C4:C4:28D10"); // GroupNames label
G:=SmallGroup(320,1328);
// by ID
G=gap.SmallGroup(320,1328);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1123,346,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations