p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.485C23, C4.712- 1+4, Q8.Q8⋊7C2, C4⋊C4.268D4, (C4×Q16)⋊25C2, (C8×D4).11C2, C8.5Q8⋊7C2, (C2×D4).241D4, C8.75(C4○D4), C2.55(Q8○D8), C8.18D4⋊23C2, C4⋊C8.320C22, C4⋊C4.241C23, (C2×C8).196C23, (C4×C8).120C22, (C2×C4).528C24, C22⋊C4.112D4, C23.114(C2×D4), C2.81(D4⋊6D4), (C4×D4).341C22, C22.10(C4○D8), C23.25D4⋊9C2, C23.20D4⋊8C2, (C4×Q8).171C22, (C2×Q8).234C23, C2.D8.194C22, C4.Q8.108C22, C22⋊Q8.97C22, C23.47D4⋊34C2, C22⋊C8.207C22, (C22×C8).196C22, (C2×Q16).138C22, C22.788(C22×D4), C42.C2.45C22, (C22×C4).1160C23, Q8⋊C4.118C22, C42⋊C2.200C22, C22.46C24.2C2, (C2×C2.D8)⋊29C2, C2.66(C2×C4○D8), C22⋊C4○(C2.D8), C4.110(C2×C4○D4), (C2×C4).931(C2×D4), (C2×C4⋊C4).680C22, SmallGroup(128,2068)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.485C23
G = < a,b,c,d,e | a4=b4=1, c2=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: 280 in 171 conjugacy classes, 88 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C22×C8, C2×Q16, C2×C2.D8, C23.25D4, C8×D4, C4×Q16, C8.18D4, Q8.Q8, C23.47D4, C23.20D4, C8.5Q8, C22.46C24, C42.485C23
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, Q8○D8, C42.485C23
►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 18 21 25)(2 19 22 26)(3 20 23 27)(4 17 24 28)(5 14 61 11)(6 15 62 12)(7 16 63 9)(8 13 64 10)(29 39 42 34)(30 40 43 35)(31 37 44 36)(32 38 41 33)(45 52 58 53)(46 49 59 54)(47 50 60 55)(48 51 57 56)
(1 54 23 51)(2 52 24 55)(3 56 21 49)(4 50 22 53)(5 31 63 42)(6 43 64 32)(7 29 61 44)(8 41 62 30)(9 39 14 36)(10 33 15 40)(11 37 16 34)(12 35 13 38)(17 47 26 58)(18 59 27 48)(19 45 28 60)(20 57 25 46)
(1 35 23 38)(2 36 24 39)(3 33 21 40)(4 34 22 37)(5 52 63 55)(6 49 64 56)(7 50 61 53)(8 51 62 54)(9 60 14 45)(10 57 15 46)(11 58 16 47)(12 59 13 48)(17 42 26 31)(18 43 27 32)(19 44 28 29)(20 41 25 30)
(1 25 21 18)(2 17 22 28)(3 27 23 20)(4 19 24 26)(5 11 61 14)(6 13 62 10)(7 9 63 16)(8 15 64 12)(29 36 42 37)(30 40 43 35)(31 34 44 39)(32 38 41 33)(45 52 58 53)(46 56 59 51)(47 50 60 55)(48 54 57 49)
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), (1,18,21,25)(2,19,22,26)(3,20,23,27)(4,17,24,28)(5,14,61,11)(6,15,62,12)(7,16,63,9)(8,13,64,10)(29,39,42,34)(30,40,43,35)(31,37,44,36)(32,38,41,33)(45,52,58,53)(46,49,59,54)(47,50,60,55)(48,51,57,56), (1,54,23,51)(2,52,24,55)(3,56,21,49)(4,50,22,53)(5,31,63,42)(6,43,64,32)(7,29,61,44)(8,41,62,30)(9,39,14,36)(10,33,15,40)(11,37,16,34)(12,35,13,38)(17,47,26,58)(18,59,27,48)(19,45,28,60)(20,57,25,46), (1,35,23,38)(2,36,24,39)(3,33,21,40)(4,34,22,37)(5,52,63,55)(6,49,64,56)(7,50,61,53)(8,51,62,54)(9,60,14,45)(10,57,15,46)(11,58,16,47)(12,59,13,48)(17,42,26,31)(18,43,27,32)(19,44,28,29)(20,41,25,30), (1,25,21,18)(2,17,22,28)(3,27,23,20)(4,19,24,26)(5,11,61,14)(6,13,62,10)(7,9,63,16)(8,15,64,12)(29,36,42,37)(30,40,43,35)(31,34,44,39)(32,38,41,33)(45,52,58,53)(46,56,59,51)(47,50,60,55)(48,54,57,49)>;
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), (1,18,21,25)(2,19,22,26)(3,20,23,27)(4,17,24,28)(5,14,61,11)(6,15,62,12)(7,16,63,9)(8,13,64,10)(29,39,42,34)(30,40,43,35)(31,37,44,36)(32,38,41,33)(45,52,58,53)(46,49,59,54)(47,50,60,55)(48,51,57,56), (1,54,23,51)(2,52,24,55)(3,56,21,49)(4,50,22,53)(5,31,63,42)(6,43,64,32)(7,29,61,44)(8,41,62,30)(9,39,14,36)(10,33,15,40)(11,37,16,34)(12,35,13,38)(17,47,26,58)(18,59,27,48)(19,45,28,60)(20,57,25,46), (1,35,23,38)(2,36,24,39)(3,33,21,40)(4,34,22,37)(5,52,63,55)(6,49,64,56)(7,50,61,53)(8,51,62,54)(9,60,14,45)(10,57,15,46)(11,58,16,47)(12,59,13,48)(17,42,26,31)(18,43,27,32)(19,44,28,29)(20,41,25,30), (1,25,21,18)(2,17,22,28)(3,27,23,20)(4,19,24,26)(5,11,61,14)(6,13,62,10)(7,9,63,16)(8,15,64,12)(29,36,42,37)(30,40,43,35)(31,34,44,39)(32,38,41,33)(45,52,58,53)(46,56,59,51)(47,50,60,55)(48,54,57,49) );
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)], [(1,18,21,25),(2,19,22,26),(3,20,23,27),(4,17,24,28),(5,14,61,11),(6,15,62,12),(7,16,63,9),(8,13,64,10),(29,39,42,34),(30,40,43,35),(31,37,44,36),(32,38,41,33),(45,52,58,53),(46,49,59,54),(47,50,60,55),(48,51,57,56)], [(1,54,23,51),(2,52,24,55),(3,56,21,49),(4,50,22,53),(5,31,63,42),(6,43,64,32),(7,29,61,44),(8,41,62,30),(9,39,14,36),(10,33,15,40),(11,37,16,34),(12,35,13,38),(17,47,26,58),(18,59,27,48),(19,45,28,60),(20,57,25,46)], [(1,35,23,38),(2,36,24,39),(3,33,21,40),(4,34,22,37),(5,52,63,55),(6,49,64,56),(7,50,61,53),(8,51,62,54),(9,60,14,45),(10,57,15,46),(11,58,16,47),(12,59,13,48),(17,42,26,31),(18,43,27,32),(19,44,28,29),(20,41,25,30)], [(1,25,21,18),(2,17,22,28),(3,27,23,20),(4,19,24,26),(5,11,61,14),(6,13,62,10),(7,9,63,16),(8,15,64,12),(29,36,42,37),(30,40,43,35),(31,34,44,39),(32,38,41,33),(45,52,58,53),(46,56,59,51),(47,50,60,55),(48,54,57,49)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | ··· | 4L | 4M | ··· | 4R | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 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 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2- 1+4 | Q8○D8 |
| kernel | C42.485C23 | C2×C2.D8 | C23.25D4 | C8×D4 | C4×Q16 | C8.18D4 | Q8.Q8 | C23.47D4 | C23.20D4 | C8.5Q8 | C22.46C24 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.485C23 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,1,0,0,15,1],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,15,0,0,8,0,0,0,0,0,13,0,0,0,9,4],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,16,0,0,0,16] >;
C42.485C23 in GAP, Magma, Sage, TeX
C_4^2._{485}C_2^3 % in TeX
G:=Group("C4^2.485C2^3"); // GroupNames label
G:=SmallGroup(128,2068);
// by ID
G=gap.SmallGroup(128,2068);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,100,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^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