metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊26D10, (C4×D5)⋊13D4, (C2×Q8)⋊15D10, C4.186(D4×D5), C22⋊Q8⋊26D5, D10⋊4(C4○D4), C20⋊7D4⋊35C2, C4⋊2D20⋊23C2, D10.76(C2×D4), C20.231(C2×D4), C22⋊D20⋊15C2, D20⋊8C4⋊24C2, (Q8×C10)⋊6C22, D10⋊3Q8⋊13C2, (C2×D20)⋊24C22, C4⋊Dic5⋊35C22, C22⋊C4.56D10, C10.73(C22×D4), Dic5⋊4D4⋊14C2, (C2×C10).171C24, (C2×C20).598C23, Dic5.119(C2×D4), C5⋊5(C22.19C24), C22⋊1(Q8⋊2D5), (C4×Dic5)⋊27C22, (C22×C4).372D10, D10.13D4⋊15C2, D10⋊C4⋊19C22, C10.D4⋊17C22, (C2×Dic5).86C23, C22.192(C23×D5), C23.188(C22×D5), (C22×C10).199C23, (C22×C20).251C22, (C22×D5).203C23, (C23×D5).121C22, (C22×Dic5).248C22, C2.46(C2×D4×D5), (D5×C22×C4)⋊5C2, C2.48(D5×C4○D4), (C2×C4×D5)⋊17C22, C4⋊C4⋊7D5⋊24C2, (C5×C22⋊Q8)⋊7C2, (C2×C10)⋊6(C4○D4), (C5×C4⋊C4)⋊18C22, (C2×Q8⋊2D5)⋊5C2, C10.160(C2×C4○D4), C2.16(C2×Q8⋊2D5), (C2×C4).46(C22×D5), (C2×C5⋊D4).38C22, (C5×C22⋊C4).26C22, SmallGroup(320,1299)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1294 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], D4 [×14], Q8 [×2], C23, C23 [×10], D5 [×6], C10 [×3], C10 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×11], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×5], D10 [×4], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×2], C23×C4, C2×C4○D4, C4×D5 [×4], C4×D5 [×10], D20 [×10], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C22.19C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5 [×4], C2×C4×D5 [×2], C2×C4×D5 [×4], C2×D20, C2×D20 [×4], Q8⋊2D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C22×C20, Q8×C10, C23×D5, Dic5⋊4D4 [×2], C22⋊D20 [×2], C4⋊C4⋊7D5, D20⋊8C4 [×2], D10.13D4 [×2], C4⋊2D20, C20⋊7D4, D10⋊3Q8, C5×C22⋊Q8, D5×C22×C4, C2×Q8⋊2D5, C4⋊C4⋊26D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.19C24, D4×D5 [×2], Q8⋊2D5 [×2], C23×D5, C2×D4×D5, C2×Q8⋊2D5, D5×C4○D4, C4⋊C4⋊26D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >
►On 80 points
(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 75 26 80)(22 76 27 71)(23 77 28 72)(24 78 29 73)(25 79 30 74)(31 67 36 62)(32 68 37 63)(33 69 38 64)(34 70 39 65)(35 61 40 66)
(1 33 6 29)(2 25 7 39)(3 35 8 21)(4 27 9 31)(5 37 10 23)(11 36 16 22)(12 28 17 32)(13 38 18 24)(14 30 19 34)(15 40 20 26)(41 67 46 71)(42 77 47 63)(43 69 48 73)(44 79 49 65)(45 61 50 75)(51 62 56 76)(52 72 57 68)(53 64 58 78)(54 74 59 70)(55 66 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 10)(2 9)(3 8)(4 7)(5 6)(11 14)(12 13)(15 20)(16 19)(17 18)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 44)(42 43)(45 50)(46 49)(47 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)
G:=sub<Sym(80)| (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,75,26,80)(22,76,27,71)(23,77,28,72)(24,78,29,73)(25,79,30,74)(31,67,36,62)(32,68,37,63)(33,69,38,64)(34,70,39,65)(35,61,40,66), (1,33,6,29)(2,25,7,39)(3,35,8,21)(4,27,9,31)(5,37,10,23)(11,36,16,22)(12,28,17,32)(13,38,18,24)(14,30,19,34)(15,40,20,26)(41,67,46,71)(42,77,47,63)(43,69,48,73)(44,79,49,65)(45,61,50,75)(51,62,56,76)(52,72,57,68)(53,64,58,78)(54,74,59,70)(55,66,60,80), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)>;
G:=Group( (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,75,26,80)(22,76,27,71)(23,77,28,72)(24,78,29,73)(25,79,30,74)(31,67,36,62)(32,68,37,63)(33,69,38,64)(34,70,39,65)(35,61,40,66), (1,33,6,29)(2,25,7,39)(3,35,8,21)(4,27,9,31)(5,37,10,23)(11,36,16,22)(12,28,17,32)(13,38,18,24)(14,30,19,34)(15,40,20,26)(41,67,46,71)(42,77,47,63)(43,69,48,73)(44,79,49,65)(45,61,50,75)(51,62,56,76)(52,72,57,68)(53,64,58,78)(54,74,59,70)(55,66,60,80), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71) );
G=PermutationGroup([(1,53,13,43),(2,54,14,44),(3,55,15,45),(4,56,16,46),(5,57,17,47),(6,58,18,48),(7,59,19,49),(8,60,20,50),(9,51,11,41),(10,52,12,42),(21,75,26,80),(22,76,27,71),(23,77,28,72),(24,78,29,73),(25,79,30,74),(31,67,36,62),(32,68,37,63),(33,69,38,64),(34,70,39,65),(35,61,40,66)], [(1,33,6,29),(2,25,7,39),(3,35,8,21),(4,27,9,31),(5,37,10,23),(11,36,16,22),(12,28,17,32),(13,38,18,24),(14,30,19,34),(15,40,20,26),(41,67,46,71),(42,77,47,63),(43,69,48,73),(44,79,49,65),(45,61,50,75),(51,62,56,76),(52,72,57,68),(53,64,58,78),(54,74,59,70),(55,66,60,80)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13),(15,20),(16,19),(17,18),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,44),(42,43),(45,50),(46,49),(47,48),(51,54),(52,53),(55,60),(56,59),(57,58),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 7 | 0 | 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 | 1 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,32,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
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 | 20A | ··· | 20H | 20I | ··· | 20P |
| 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 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D4×D5 | Q8⋊2D5 | D5×C4○D4 |
| kernel | C4⋊C4⋊26D10 | Dic5⋊4D4 | C22⋊D20 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | C20⋊7D4 | D10⋊3Q8 | C5×C22⋊Q8 | D5×C22×C4 | C2×Q8⋊2D5 | C4×D5 | C22⋊Q8 | D10 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 6 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{26}D_{10} % in TeX
G:=Group("C4:C4:26D10"); // GroupNames label
G:=SmallGroup(320,1299);
// by ID
G=gap.SmallGroup(320,1299);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1123,794,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations