metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊C4.1F5, C22.F5⋊1C4, C23.6(C2×F5), C22.7(C4×F5), (C2×C10).2C42, (C2×Dic5).9Q8, C22.9(C4⋊F5), C23.D5.1C4, Dic5.2(C4⋊C4), C5⋊1(M4(2)⋊4C4), (C2×Dic5).254D4, C2.6(D10.3Q8), C22.17(C22⋊F5), C10.4(C2.C42), Dic5.29(C22⋊C4), C23.11D10.4C2, (C22×Dic5).170C22, (C2×C5⋊C8)⋊1C4, (C2×C10).2(C4⋊C4), (C5×C22⋊C4).1C4, (C2×C22.F5).1C2, (C22×C10).11(C2×C4), (C2×Dic5).41(C2×C4), (C2×C10).17(C22⋊C4), SmallGroup(320,205)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4.F5
G = < a,b,c,d,e | a2=b2=c5=e4=1, d4=b, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, dcd-1=c3, ce=ec, ede-1=abd >
Subgroups: 322 in 90 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C42⋊C2, C2×M4(2), C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C22×C10, M4(2)⋊4C4, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C2×C5⋊C8, C2×C5⋊C8, C22.F5, C22.F5, C22×Dic5, C23.11D10, C2×C22.F5, C22⋊C4.F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C2×F5, M4(2)⋊4C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C22⋊C4.F5
►On 80 points
Generators in S80
(1 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)(41 62)(42 59)(43 64)(44 61)(45 58)(46 63)(47 60)(48 57)(65 76)(66 73)(67 78)(68 75)(69 80)(70 77)(71 74)(72 79)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)
(1 78 47 33 19)(2 34 79 20 48)(3 21 35 41 80)(4 42 22 73 36)(5 74 43 37 23)(6 38 75 24 44)(7 17 39 45 76)(8 46 18 77 40)(9 25 68 49 61)(10 50 26 62 69)(11 63 51 70 27)(12 71 64 28 52)(13 29 72 53 57)(14 54 30 58 65)(15 59 55 66 31)(16 67 60 32 56)
(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 14 5 10)(2 8)(3 12 7 16)(4 6)(9 11)(13 15)(17 67 21 71)(18 79)(19 65 23 69)(20 77)(22 75)(24 73)(25 63)(26 47 30 43)(27 61)(28 45 32 41)(29 59)(31 57)(33 58 37 62)(34 46)(35 64 39 60)(36 44)(38 42)(40 48)(49 70)(50 78 54 74)(51 68)(52 76 56 80)(53 66)(55 72)
G:=sub<Sym(80)| (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(41,62)(42,59)(43,64)(44,61)(45,58)(46,63)(47,60)(48,57)(65,76)(66,73)(67,78)(68,75)(69,80)(70,77)(71,74)(72,79), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (1,78,47,33,19)(2,34,79,20,48)(3,21,35,41,80)(4,42,22,73,36)(5,74,43,37,23)(6,38,75,24,44)(7,17,39,45,76)(8,46,18,77,40)(9,25,68,49,61)(10,50,26,62,69)(11,63,51,70,27)(12,71,64,28,52)(13,29,72,53,57)(14,54,30,58,65)(15,59,55,66,31)(16,67,60,32,56), (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), (1,14,5,10)(2,8)(3,12,7,16)(4,6)(9,11)(13,15)(17,67,21,71)(18,79)(19,65,23,69)(20,77)(22,75)(24,73)(25,63)(26,47,30,43)(27,61)(28,45,32,41)(29,59)(31,57)(33,58,37,62)(34,46)(35,64,39,60)(36,44)(38,42)(40,48)(49,70)(50,78,54,74)(51,68)(52,76,56,80)(53,66)(55,72)>;
G:=Group( (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(41,62)(42,59)(43,64)(44,61)(45,58)(46,63)(47,60)(48,57)(65,76)(66,73)(67,78)(68,75)(69,80)(70,77)(71,74)(72,79), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (1,78,47,33,19)(2,34,79,20,48)(3,21,35,41,80)(4,42,22,73,36)(5,74,43,37,23)(6,38,75,24,44)(7,17,39,45,76)(8,46,18,77,40)(9,25,68,49,61)(10,50,26,62,69)(11,63,51,70,27)(12,71,64,28,52)(13,29,72,53,57)(14,54,30,58,65)(15,59,55,66,31)(16,67,60,32,56), (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), (1,14,5,10)(2,8)(3,12,7,16)(4,6)(9,11)(13,15)(17,67,21,71)(18,79)(19,65,23,69)(20,77)(22,75)(24,73)(25,63)(26,47,30,43)(27,61)(28,45,32,41)(29,59)(31,57)(33,58,37,62)(34,46)(35,64,39,60)(36,44)(38,42)(40,48)(49,70)(50,78,54,74)(51,68)(52,76,56,80)(53,66)(55,72) );
G=PermutationGroup([[(1,16),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11),(17,54),(18,51),(19,56),(20,53),(21,50),(22,55),(23,52),(24,49),(25,38),(26,35),(27,40),(28,37),(29,34),(30,39),(31,36),(32,33),(41,62),(42,59),(43,64),(44,61),(45,58),(46,63),(47,60),(48,57),(65,76),(66,73),(67,78),(68,75),(69,80),(70,77),(71,74),(72,79)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80)], [(1,78,47,33,19),(2,34,79,20,48),(3,21,35,41,80),(4,42,22,73,36),(5,74,43,37,23),(6,38,75,24,44),(7,17,39,45,76),(8,46,18,77,40),(9,25,68,49,61),(10,50,26,62,69),(11,63,51,70,27),(12,71,64,28,52),(13,29,72,53,57),(14,54,30,58,65),(15,59,55,66,31),(16,67,60,32,56)], [(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)], [(1,14,5,10),(2,8),(3,12,7,16),(4,6),(9,11),(13,15),(17,67,21,71),(18,79),(19,65,23,69),(20,77),(22,75),(24,73),(25,63),(26,47,30,43),(27,61),(28,45,32,41),(29,59),(31,57),(33,58,37,62),(34,46),(35,64,39,60),(36,44),(38,42),(40,48),(49,70),(50,78,54,74),(51,68),(52,76,56,80),(53,66),(55,72)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 10 | 20 | 20 | 4 | 20 | ··· | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | F5 | C2×F5 | M4(2)⋊4C4 | C4×F5 | C4⋊F5 | C22⋊F5 | C22⋊C4.F5 |
| kernel | C22⋊C4.F5 | C23.11D10 | C2×C22.F5 | C23.D5 | C5×C22⋊C4 | C2×C5⋊C8 | C22.F5 | C2×Dic5 | C2×Dic5 | C22⋊C4 | C23 | C5 | C22 | C22 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of C22⋊C4.F5 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 39 | 40 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 0 |
| 9 | 9 | 2 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 37 | 37 | 32 | 0 | 0 | 0 | 0 | 0 |
| 39 | 40 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 7 | 0 | 0 |
| 32 | 32 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,16,1,21,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,33,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,34,34,0,0,0,0,0,0,6,0],[9,0,37,39,0,0,0,0,9,0,37,40,0,0,0,0,2,0,32,16,0,0,0,0,32,9,0,0,0,0,0,0,0,0,0,0,0,0,34,34,0,0,0,0,0,0,1,7,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[32,0,0,0,0,0,0,0,32,9,0,0,0,0,0,0,0,0,32,16,0,0,0,0,1,0,36,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
C22⋊C4.F5 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4.F_5
% in TeX
G:=Group("C2^2:C4.F5"); // GroupNames label
G:=SmallGroup(320,205);
// by ID
G=gap.SmallGroup(320,205);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,851,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^5=e^4=1,d^4=b,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^3,c*e=e*c,e*d*e^-1=a*b*d>;
// generators/relations