p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.470C23, C4.502+ 1+4, (C8×D4)⋊23C2, C4⋊D8⋊14C2, C8⋊7D4⋊14C2, C4⋊C4.411D4, Q8⋊6D4⋊7C2, D4⋊D4⋊12C2, C4.Q16⋊13C2, (C4×SD16)⋊42C2, (C2×D4).240D4, C2.46(D4○D8), C4.73(C4○D8), C4.4D8⋊30C2, C4⋊C8.298C22, C4⋊C4.405C23, (C2×C4).497C24, (C2×C8).351C23, (C4×C8).275C22, Q8.20(C4○D4), C22⋊C4.109D4, (C2×D8).37C22, C23.113(C2×D4), C4⋊Q8.146C22, C2.D8.57C22, (C2×D4).227C23, (C4×D4).338C22, C4⋊D4.77C22, C23.19D4⋊7C2, C4⋊1D4.85C22, (C2×Q8).395C23, (C4×Q8).153C22, C2.133(D4⋊5D4), C4.Q8.167C22, C23.24D4⋊13C2, C22⋊C8.206C22, (C22×C8).163C22, C22.757(C22×D4), D4⋊C4.120C22, (C22×C4).1141C23, C22.49C24⋊4C2, Q8⋊C4.159C22, (C2×SD16).158C22, C42⋊C2.185C22, C4⋊C4○(Q8⋊C4), C2.65(C2×C4○D8), C4.222(C2×C4○D4), (C2×C4).926(C2×D4), (C2×C4○D4).203C22, SmallGroup(128,2037)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.470C23
G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, de=ed >
Subgroups: 440 in 204 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C2×SD16, C2×C4○D4, C23.24D4, C8×D4, C4×SD16, D4⋊D4, C4⋊D8, C8⋊7D4, C4.Q16, C23.19D4, C4.4D8, Q8⋊6D4, C22.49C24, C42.470C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C4○D8, D4○D8, C42.470C23
►On 64 points
(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)
(1 42 49 12)(2 43 50 9)(3 44 51 10)(4 41 52 11)(5 31 40 56)(6 32 37 53)(7 29 38 54)(8 30 39 55)(13 20 46 21)(14 17 47 22)(15 18 48 23)(16 19 45 24)(25 59 33 62)(26 60 34 63)(27 57 35 64)(28 58 36 61)
(1 6 51 39)(2 5 52 38)(3 8 49 37)(4 7 50 40)(9 31 41 54)(10 30 42 53)(11 29 43 56)(12 32 44 55)(13 28 48 34)(14 27 45 33)(15 26 46 36)(16 25 47 35)(17 64 24 59)(18 63 21 58)(19 62 22 57)(20 61 23 60)
(1 21 49 20)(2 22 50 17)(3 23 51 18)(4 24 52 19)(5 27 40 35)(6 28 37 36)(7 25 38 33)(8 26 39 34)(9 14 43 47)(10 15 44 48)(11 16 41 45)(12 13 42 46)(29 62 54 59)(30 63 55 60)(31 64 56 57)(32 61 53 58)
(1 24 49 19)(2 23 50 18)(3 22 51 17)(4 21 52 20)(5 58 40 61)(6 57 37 64)(7 60 38 63)(8 59 39 62)(9 48 43 15)(10 47 44 14)(11 46 41 13)(12 45 42 16)(25 30 33 55)(26 29 34 54)(27 32 35 53)(28 31 36 56)
G:=sub<Sym(64)| (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), (1,42,49,12)(2,43,50,9)(3,44,51,10)(4,41,52,11)(5,31,40,56)(6,32,37,53)(7,29,38,54)(8,30,39,55)(13,20,46,21)(14,17,47,22)(15,18,48,23)(16,19,45,24)(25,59,33,62)(26,60,34,63)(27,57,35,64)(28,58,36,61), (1,6,51,39)(2,5,52,38)(3,8,49,37)(4,7,50,40)(9,31,41,54)(10,30,42,53)(11,29,43,56)(12,32,44,55)(13,28,48,34)(14,27,45,33)(15,26,46,36)(16,25,47,35)(17,64,24,59)(18,63,21,58)(19,62,22,57)(20,61,23,60), (1,21,49,20)(2,22,50,17)(3,23,51,18)(4,24,52,19)(5,27,40,35)(6,28,37,36)(7,25,38,33)(8,26,39,34)(9,14,43,47)(10,15,44,48)(11,16,41,45)(12,13,42,46)(29,62,54,59)(30,63,55,60)(31,64,56,57)(32,61,53,58), (1,24,49,19)(2,23,50,18)(3,22,51,17)(4,21,52,20)(5,58,40,61)(6,57,37,64)(7,60,38,63)(8,59,39,62)(9,48,43,15)(10,47,44,14)(11,46,41,13)(12,45,42,16)(25,30,33,55)(26,29,34,54)(27,32,35,53)(28,31,36,56)>;
G:=Group( (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), (1,42,49,12)(2,43,50,9)(3,44,51,10)(4,41,52,11)(5,31,40,56)(6,32,37,53)(7,29,38,54)(8,30,39,55)(13,20,46,21)(14,17,47,22)(15,18,48,23)(16,19,45,24)(25,59,33,62)(26,60,34,63)(27,57,35,64)(28,58,36,61), (1,6,51,39)(2,5,52,38)(3,8,49,37)(4,7,50,40)(9,31,41,54)(10,30,42,53)(11,29,43,56)(12,32,44,55)(13,28,48,34)(14,27,45,33)(15,26,46,36)(16,25,47,35)(17,64,24,59)(18,63,21,58)(19,62,22,57)(20,61,23,60), (1,21,49,20)(2,22,50,17)(3,23,51,18)(4,24,52,19)(5,27,40,35)(6,28,37,36)(7,25,38,33)(8,26,39,34)(9,14,43,47)(10,15,44,48)(11,16,41,45)(12,13,42,46)(29,62,54,59)(30,63,55,60)(31,64,56,57)(32,61,53,58), (1,24,49,19)(2,23,50,18)(3,22,51,17)(4,21,52,20)(5,58,40,61)(6,57,37,64)(7,60,38,63)(8,59,39,62)(9,48,43,15)(10,47,44,14)(11,46,41,13)(12,45,42,16)(25,30,33,55)(26,29,34,54)(27,32,35,53)(28,31,36,56) );
G=PermutationGroup([[(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)], [(1,42,49,12),(2,43,50,9),(3,44,51,10),(4,41,52,11),(5,31,40,56),(6,32,37,53),(7,29,38,54),(8,30,39,55),(13,20,46,21),(14,17,47,22),(15,18,48,23),(16,19,45,24),(25,59,33,62),(26,60,34,63),(27,57,35,64),(28,58,36,61)], [(1,6,51,39),(2,5,52,38),(3,8,49,37),(4,7,50,40),(9,31,41,54),(10,30,42,53),(11,29,43,56),(12,32,44,55),(13,28,48,34),(14,27,45,33),(15,26,46,36),(16,25,47,35),(17,64,24,59),(18,63,21,58),(19,62,22,57),(20,61,23,60)], [(1,21,49,20),(2,22,50,17),(3,23,51,18),(4,24,52,19),(5,27,40,35),(6,28,37,36),(7,25,38,33),(8,26,39,34),(9,14,43,47),(10,15,44,48),(11,16,41,45),(12,13,42,46),(29,62,54,59),(30,63,55,60),(31,64,56,57),(32,61,53,58)], [(1,24,49,19),(2,23,50,18),(3,22,51,17),(4,21,52,20),(5,58,40,61),(6,57,37,64),(7,60,38,63),(8,59,39,62),(9,48,43,15),(10,47,44,14),(11,46,41,13),(12,45,42,16),(25,30,33,55),(26,29,34,54),(27,32,35,53),(28,31,36,56)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | ··· | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2+ 1+4 | D4○D8 |
| kernel | C42.470C23 | C23.24D4 | C8×D4 | C4×SD16 | D4⋊D4 | C4⋊D8 | C8⋊7D4 | C4.Q16 | C23.19D4 | C4.4D8 | Q8⋊6D4 | C22.49C24 | C22⋊C4 | C4⋊C4 | C2×D4 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.470C23 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 7 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 4 | 4 |
| 4 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,16,0,0,2,16],[1,1,0,0,15,16,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,7,0,0,0,0,0,13,4,0,0,0,4],[4,4,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,2,16] >;
C42.470C23 in GAP, Magma, Sage, TeX
C_4^2._{470}C_2^3 % in TeX
G:=Group("C4^2.470C2^3"); // GroupNames label
G:=SmallGroup(128,2037);
// by ID
G=gap.SmallGroup(128,2037);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,352,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2*b^2,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*c,d*e=e*d>;
// generators/relations