metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×F5)⋊1D4, C20⋊(C22⋊C4), (C2×D4)⋊10F5, C2.30(D4×F5), (D4×C10)⋊10C4, (C2×D20)⋊10C4, (C4×D5).39D4, C10.30(C4×D4), C4⋊1(C22⋊F5), D10.96(C2×D4), Dic5⋊(C22⋊C4), C23.16(C2×F5), D5.2(C4⋊1D4), D5.5(C4⋊D4), C5⋊(C24.3C22), D10.49(C4○D4), D5.3(C4.4D4), (C22×F5).9C22, C22.96(C22×F5), (C23×D5).89C22, (C22×D5).279C23, (C2×C4×F5)⋊2C2, (C2×C4⋊F5)⋊2C2, (C2×C5⋊D4)⋊5C4, (C2×D4×D5).16C2, (C2×C22⋊F5)⋊4C2, (C2×C4).83(C2×F5), (C2×C20).58(C2×C4), C2.25(C2×C22⋊F5), C10.24(C2×C22⋊C4), (C2×C4×D5).203C22, (C22×C10).30(C2×C4), (C2×C10).83(C22×C4), (C2×Dic5).75(C2×C4), (C22×D5).57(C2×C4), SmallGroup(320,1118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.(D4×F5)
G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ab=ba, ece-1=ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ede-1=d3 >
Subgroups: 1258 in 258 conjugacy classes, 64 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, D5, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, F5, D10, D10, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C2×F5, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C24.3C22, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C22×F5, C23×D5, C2×C4×F5, C2×C4⋊F5, C2×C22⋊F5, C2×D4×D5, C2.(D4×F5)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, F5, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×F5, C24.3C22, C22⋊F5, C22×F5, D4×F5, C2×C22⋊F5, C2.(D4×F5)
►On 80 points
Generators in S80
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(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 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(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 42)(2 44 5 45)(3 41 4 43)(6 46 7 48)(8 50 10 49)(9 47)(11 51 12 53)(13 55 15 54)(14 52)(16 56 17 58)(18 60 20 59)(19 57)(21 61 22 63)(23 65 25 64)(24 62)(26 66 27 68)(28 70 30 69)(29 67)(31 71 32 73)(33 75 35 74)(34 72)(36 76 37 78)(38 80 40 79)(39 77)
G:=sub<Sym(80)| (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(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), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (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), (1,42)(2,44,5,45)(3,41,4,43)(6,46,7,48)(8,50,10,49)(9,47)(11,51,12,53)(13,55,15,54)(14,52)(16,56,17,58)(18,60,20,59)(19,57)(21,61,22,63)(23,65,25,64)(24,62)(26,66,27,68)(28,70,30,69)(29,67)(31,71,32,73)(33,75,35,74)(34,72)(36,76,37,78)(38,80,40,79)(39,77)>;
G:=Group( (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(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), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (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), (1,42)(2,44,5,45)(3,41,4,43)(6,46,7,48)(8,50,10,49)(9,47)(11,51,12,53)(13,55,15,54)(14,52)(16,56,17,58)(18,60,20,59)(19,57)(21,61,22,63)(23,65,25,64)(24,62)(26,66,27,68)(28,70,30,69)(29,67)(31,71,32,73)(33,75,35,74)(34,72)(36,76,37,78)(38,80,40,79)(39,77) );
G=PermutationGroup([[(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(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)], [(1,29,9,24),(2,30,10,25),(3,26,6,21),(4,27,7,22),(5,28,8,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(1,24),(2,25),(3,21),(4,22),(5,23),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70)], [(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)], [(1,42),(2,44,5,45),(3,41,4,43),(6,46,7,48),(8,50,10,49),(9,47),(11,51,12,53),(13,55,15,54),(14,52),(16,56,17,58),(18,60,20,59),(19,57),(21,61,22,63),(23,65,25,64),(24,62),(26,66,27,68),(28,70,30,69),(29,67),(31,71,32,73),(33,75,35,74),(34,72),(36,76,37,78),(38,80,40,79),(39,77)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | ··· | 4L | 4M | 4N | 4O | 4P | 5 | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4○D4 | F5 | C2×F5 | C2×F5 | C22⋊F5 | D4×F5 |
| kernel | C2.(D4×F5) | C2×C4×F5 | C2×C4⋊F5 | C2×C22⋊F5 | C2×D4×D5 | C2×D20 | C2×C5⋊D4 | D4×C10 | C4×D5 | C2×F5 | D10 | C2×D4 | C2×C4 | C23 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 4 | 2 |
Matrix representation of C2.(D4×F5) ►in GL6(𝔽41)
| 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 |
| 28 | 5 | 0 | 0 | 0 | 0 |
| 7 | 13 | 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 |
| 28 | 5 | 0 | 0 | 0 | 0 |
| 32 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 6 |
| 0 | 0 | 0 | 0 | 35 | 35 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [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],[28,7,0,0,0,0,5,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,32,0,0,0,0,5,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,35,0,0,0,0,6,35],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
C2.(D4×F5) in GAP, Magma, Sage, TeX
C_2.(D_4\times F_5)
% in TeX
G:=Group("C2.(D4xF5)"); // GroupNames label
G:=SmallGroup(320,1118);
// by ID
G=gap.SmallGroup(320,1118);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=e^4=1,a*b=b*a,e*c*e^-1=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*d*e^-1=d^3>;
// generators/relations