p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.486C23, C4.722- 1+4, (C8×D4)⋊24C2, C8⋊8D4⋊9C2, D4.Q8⋊7C2, Q8.Q8⋊8C2, C4⋊C4.269D4, C8.5Q8⋊8C2, (C4×SD16)⋊10C2, (C2×D4).242D4, C8.82(C4○D4), C4⋊C4.242C23, C4⋊C8.321C22, (C2×C8).361C23, (C4×C8).121C22, (C2×C4).529C24, C22⋊C4.113D4, C23.115(C2×D4), C2.82(D4⋊6D4), C2.D8.62C22, C2.87(D4○SD16), (C4×D4).342C22, (C2×D4).250C23, C22.13(C4○D8), C23.19D4⋊8C2, C4⋊D4.99C22, C22.D8⋊10C2, C23.48D4⋊9C2, C23.20D4⋊9C2, (C4×Q8).172C22, (C2×Q8).235C23, C4.Q8.169C22, C22⋊Q8.98C22, C23.25D4⋊10C2, C22⋊C8.208C22, (C22×C8).197C22, Q8⋊C4.17C22, C22.789(C22×D4), C42.C2.46C22, D4⋊C4.170C22, C22.46C24⋊8C2, (C22×C4).1161C23, (C2×SD16).165C22, C42⋊C2.201C22, C22.47C24.2C2, (C2×C4.Q8)⋊37C2, C2.67(C2×C4○D8), C22⋊C4○(C4.Q8), C4.111(C2×C4○D4), (C2×C4).932(C2×D4), (C2×C4⋊C4).681C22, SmallGroup(128,2069)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.486C23
G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
Subgroups: 320 in 177 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C22×C8, C2×SD16, C2×C4.Q8, C23.25D4, C8×D4, C4×SD16, C8⋊8D4, D4.Q8, Q8.Q8, C22.D8, C23.19D4, C23.48D4, C23.20D4, C8.5Q8, C22.46C24, C22.47C24, C42.486C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, C2×C4○D8, D4○SD16, C42.486C23
►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 60 34 23)(2 57 35 24)(3 58 36 21)(4 59 33 22)(5 12 47 26)(6 9 48 27)(7 10 45 28)(8 11 46 25)(13 18 50 53)(14 19 51 54)(15 20 52 55)(16 17 49 56)(29 61 39 44)(30 62 40 41)(31 63 37 42)(32 64 38 43)
(1 49 3 51)(2 15 4 13)(5 30 7 32)(6 39 8 37)(9 61 11 63)(10 43 12 41)(14 34 16 36)(17 58 19 60)(18 24 20 22)(21 54 23 56)(25 42 27 44)(26 62 28 64)(29 46 31 48)(33 50 35 52)(38 47 40 45)(53 57 55 59)
(1 26 36 10)(2 27 33 11)(3 28 34 12)(4 25 35 9)(5 58 45 23)(6 59 46 24)(7 60 47 21)(8 57 48 22)(13 37 52 29)(14 38 49 30)(15 39 50 31)(16 40 51 32)(17 62 54 43)(18 63 55 44)(19 64 56 41)(20 61 53 42)
(1 16 34 49)(2 52 35 15)(3 14 36 51)(4 50 33 13)(5 41 47 62)(6 61 48 44)(7 43 45 64)(8 63 46 42)(9 39 27 29)(10 32 28 38)(11 37 25 31)(12 30 26 40)(17 23 56 60)(18 59 53 22)(19 21 54 58)(20 57 55 24)
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), (1,60,34,23)(2,57,35,24)(3,58,36,21)(4,59,33,22)(5,12,47,26)(6,9,48,27)(7,10,45,28)(8,11,46,25)(13,18,50,53)(14,19,51,54)(15,20,52,55)(16,17,49,56)(29,61,39,44)(30,62,40,41)(31,63,37,42)(32,64,38,43), (1,49,3,51)(2,15,4,13)(5,30,7,32)(6,39,8,37)(9,61,11,63)(10,43,12,41)(14,34,16,36)(17,58,19,60)(18,24,20,22)(21,54,23,56)(25,42,27,44)(26,62,28,64)(29,46,31,48)(33,50,35,52)(38,47,40,45)(53,57,55,59), (1,26,36,10)(2,27,33,11)(3,28,34,12)(4,25,35,9)(5,58,45,23)(6,59,46,24)(7,60,47,21)(8,57,48,22)(13,37,52,29)(14,38,49,30)(15,39,50,31)(16,40,51,32)(17,62,54,43)(18,63,55,44)(19,64,56,41)(20,61,53,42), (1,16,34,49)(2,52,35,15)(3,14,36,51)(4,50,33,13)(5,41,47,62)(6,61,48,44)(7,43,45,64)(8,63,46,42)(9,39,27,29)(10,32,28,38)(11,37,25,31)(12,30,26,40)(17,23,56,60)(18,59,53,22)(19,21,54,58)(20,57,55,24)>;
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), (1,60,34,23)(2,57,35,24)(3,58,36,21)(4,59,33,22)(5,12,47,26)(6,9,48,27)(7,10,45,28)(8,11,46,25)(13,18,50,53)(14,19,51,54)(15,20,52,55)(16,17,49,56)(29,61,39,44)(30,62,40,41)(31,63,37,42)(32,64,38,43), (1,49,3,51)(2,15,4,13)(5,30,7,32)(6,39,8,37)(9,61,11,63)(10,43,12,41)(14,34,16,36)(17,58,19,60)(18,24,20,22)(21,54,23,56)(25,42,27,44)(26,62,28,64)(29,46,31,48)(33,50,35,52)(38,47,40,45)(53,57,55,59), (1,26,36,10)(2,27,33,11)(3,28,34,12)(4,25,35,9)(5,58,45,23)(6,59,46,24)(7,60,47,21)(8,57,48,22)(13,37,52,29)(14,38,49,30)(15,39,50,31)(16,40,51,32)(17,62,54,43)(18,63,55,44)(19,64,56,41)(20,61,53,42), (1,16,34,49)(2,52,35,15)(3,14,36,51)(4,50,33,13)(5,41,47,62)(6,61,48,44)(7,43,45,64)(8,63,46,42)(9,39,27,29)(10,32,28,38)(11,37,25,31)(12,30,26,40)(17,23,56,60)(18,59,53,22)(19,21,54,58)(20,57,55,24) );
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)], [(1,60,34,23),(2,57,35,24),(3,58,36,21),(4,59,33,22),(5,12,47,26),(6,9,48,27),(7,10,45,28),(8,11,46,25),(13,18,50,53),(14,19,51,54),(15,20,52,55),(16,17,49,56),(29,61,39,44),(30,62,40,41),(31,63,37,42),(32,64,38,43)], [(1,49,3,51),(2,15,4,13),(5,30,7,32),(6,39,8,37),(9,61,11,63),(10,43,12,41),(14,34,16,36),(17,58,19,60),(18,24,20,22),(21,54,23,56),(25,42,27,44),(26,62,28,64),(29,46,31,48),(33,50,35,52),(38,47,40,45),(53,57,55,59)], [(1,26,36,10),(2,27,33,11),(3,28,34,12),(4,25,35,9),(5,58,45,23),(6,59,46,24),(7,60,47,21),(8,57,48,22),(13,37,52,29),(14,38,49,30),(15,39,50,31),(16,40,51,32),(17,62,54,43),(18,63,55,44),(19,64,56,41),(20,61,53,42)], [(1,16,34,49),(2,52,35,15),(3,14,36,51),(4,50,33,13),(5,41,47,62),(6,61,48,44),(7,43,45,64),(8,63,46,42),(9,39,27,29),(10,32,28,38),(11,37,25,31),(12,30,26,40),(17,23,56,60),(18,59,53,22),(19,21,54,58),(20,57,55,24)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 4M | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2- 1+4 | D4○SD16 |
| kernel | C42.486C23 | C2×C4.Q8 | C23.25D4 | C8×D4 | C4×SD16 | C8⋊8D4 | D4.Q8 | Q8.Q8 | C22.D8 | C23.19D4 | C23.48D4 | C23.20D4 | C8.5Q8 | C22.46C24 | C22.47C24 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.486C23 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 3 | 14 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[0,13,0,0,13,0,0,0,0,0,4,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,3,3,0,0,3,14],[0,13,0,0,4,0,0,0,0,0,13,0,0,0,0,13] >;
C42.486C23 in GAP, Magma, Sage, TeX
C_4^2._{486}C_2^3 % in TeX
G:=Group("C4^2.486C2^3"); // GroupNames label
G:=SmallGroup(128,2069);
// by ID
G=gap.SmallGroup(128,2069);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,100,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a^2*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,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*b^2*c,d*e=e*d>;
// generators/relations