metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊15D4, (C2×D4)⋊43D10, (C2×Q8)⋊32D10, D10⋊6(C4○D4), C20⋊2D4⋊46C2, C20.265(C2×D4), (C22×C4)⋊29D10, C23⋊D10⋊34C2, D10⋊3Q8⋊47C2, (D4×C10)⋊47C22, C4⋊Dic5⋊80C22, (Q8×C10)⋊39C22, (C2×C20).651C23, (C2×C10).314C24, (C22×C20)⋊42C22, C5⋊8(C22.19C24), (C4×Dic5)⋊60C22, C10.164(C22×D4), C23.D5⋊65C22, C10.D4⋊76C22, C22.323(C23×D5), C23.137(C22×D5), C23.18D10⋊34C2, C23.21D10⋊38C2, (C22×C10).240C23, (C2×Dic5).303C23, (C22×D5).253C23, (C23×D5).127C22, D10⋊C4.159C22, (C22×Dic5).259C22, (C2×C4○D4)⋊6D5, (D5×C22×C4)⋊7C2, (C10×C4○D4)⋊6C2, (C4×C5⋊D4)⋊60C2, (C2×C4)⋊14(C5⋊D4), C2.103(D5×C4○D4), (C2×C10).80(C2×D4), C4.100(C2×C5⋊D4), C10.215(C2×C4○D4), (C2×C4×D5).333C22, C22.23(C2×C5⋊D4), C2.37(C22×C5⋊D4), (C2×C4).639(C22×D5), (C2×C5⋊D4).148C22, SmallGroup(320,1500)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1118 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D5 [×4], C10, C10 [×2], C10 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×9], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic5 [×6], C20 [×4], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×8], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×2], C22.19C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×6], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×4], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C23×D5, C23.21D10, C4×C5⋊D4 [×4], C23.18D10 [×2], C23⋊D10 [×2], C20⋊2D4 [×2], D10⋊3Q8 [×2], D5×C22×C4, C10×C4○D4, (C2×C20)⋊15D4
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], C5⋊D4 [×4], C22×D5 [×7], C22.19C24, C2×C5⋊D4 [×6], C23×D5, D5×C4○D4 [×2], C22×C5⋊D4, (C2×C20)⋊15D4
Generators and relations
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, cac-1=ab10, ad=da, cbc-1=dbd=b9, dcd=c-1 >
►On 80 points
(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 77 21 60)(2 66 22 49)(3 75 23 58)(4 64 24 47)(5 73 25 56)(6 62 26 45)(7 71 27 54)(8 80 28 43)(9 69 29 52)(10 78 30 41)(11 67 31 50)(12 76 32 59)(13 65 33 48)(14 74 34 57)(15 63 35 46)(16 72 36 55)(17 61 37 44)(18 70 38 53)(19 79 39 42)(20 68 40 51)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 31)(22 40)(23 29)(24 38)(25 27)(26 36)(28 34)(30 32)(33 39)(35 37)(41 76)(42 65)(43 74)(44 63)(45 72)(46 61)(47 70)(48 79)(49 68)(50 77)(51 66)(52 75)(53 64)(54 73)(55 62)(56 71)(57 80)(58 69)(59 78)(60 67)
G:=sub<Sym(80)| (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), (1,77,21,60)(2,66,22,49)(3,75,23,58)(4,64,24,47)(5,73,25,56)(6,62,26,45)(7,71,27,54)(8,80,28,43)(9,69,29,52)(10,78,30,41)(11,67,31,50)(12,76,32,59)(13,65,33,48)(14,74,34,57)(15,63,35,46)(16,72,36,55)(17,61,37,44)(18,70,38,53)(19,79,39,42)(20,68,40,51), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,31)(22,40)(23,29)(24,38)(25,27)(26,36)(28,34)(30,32)(33,39)(35,37)(41,76)(42,65)(43,74)(44,63)(45,72)(46,61)(47,70)(48,79)(49,68)(50,77)(51,66)(52,75)(53,64)(54,73)(55,62)(56,71)(57,80)(58,69)(59,78)(60,67)>;
G:=Group( (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), (1,77,21,60)(2,66,22,49)(3,75,23,58)(4,64,24,47)(5,73,25,56)(6,62,26,45)(7,71,27,54)(8,80,28,43)(9,69,29,52)(10,78,30,41)(11,67,31,50)(12,76,32,59)(13,65,33,48)(14,74,34,57)(15,63,35,46)(16,72,36,55)(17,61,37,44)(18,70,38,53)(19,79,39,42)(20,68,40,51), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,31)(22,40)(23,29)(24,38)(25,27)(26,36)(28,34)(30,32)(33,39)(35,37)(41,76)(42,65)(43,74)(44,63)(45,72)(46,61)(47,70)(48,79)(49,68)(50,77)(51,66)(52,75)(53,64)(54,73)(55,62)(56,71)(57,80)(58,69)(59,78)(60,67) );
G=PermutationGroup([(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)], [(1,77,21,60),(2,66,22,49),(3,75,23,58),(4,64,24,47),(5,73,25,56),(6,62,26,45),(7,71,27,54),(8,80,28,43),(9,69,29,52),(10,78,30,41),(11,67,31,50),(12,76,32,59),(13,65,33,48),(14,74,34,57),(15,63,35,46),(16,72,36,55),(17,61,37,44),(18,70,38,53),(19,79,39,42),(20,68,40,51)], [(1,11),(2,20),(3,9),(4,18),(5,7),(6,16),(8,14),(10,12),(13,19),(15,17),(21,31),(22,40),(23,29),(24,38),(25,27),(26,36),(28,34),(30,32),(33,39),(35,37),(41,76),(42,65),(43,74),(44,63),(45,72),(46,61),(47,70),(48,79),(49,68),(50,77),(51,66),(52,75),(53,64),(54,73),(55,62),(56,71),(57,80),(58,69),(59,78),(60,67)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
| 1 | 1 | 0 | 0 |
| 5 | 6 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 20 | 3 | 0 | 0 |
| 3 | 21 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 6 | 7 | 0 | 0 |
| 36 | 35 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,40,0,0,0,40],[1,5,0,0,1,6,0,0,0,0,32,0,0,0,0,32],[20,3,0,0,3,21,0,0,0,0,40,1,0,0,39,1],[6,36,0,0,7,35,0,0,0,0,40,1,0,0,0,1] >;
68 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 | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| 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 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D5×C4○D4 |
| kernel | (C2×C20)⋊15D4 | C23.21D10 | C4×C5⋊D4 | C23.18D10 | C23⋊D10 | C20⋊2D4 | D10⋊3Q8 | D5×C22×C4 | C10×C4○D4 | C2×C20 | C2×C4○D4 | D10 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 2 | 8 | 6 | 6 | 2 | 16 | 8 |
In GAP, Magma, Sage, TeX
(C_2\times C_{20})\rtimes_{15}D_4 % in TeX
G:=Group("(C2xC20):15D4"); // GroupNames label
G:=SmallGroup(320,1500);
// by ID
G=gap.SmallGroup(320,1500);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^10,a*d=d*a,c*b*c^-1=d*b*d=b^9,d*c*d=c^-1>;
// generators/relations