p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.489C23, C4.752- 1+4, (C8×D4)⋊26C2, C8⋊8D4⋊10C2, C4⋊C4.412D4, C8⋊3Q8⋊26C2, C4.Q16⋊15C2, D4⋊Q8⋊15C2, (C4×SD16)⋊44C2, (C2×D4).244D4, C4.74(C4○D8), C8.83(C4○D4), C4⋊C8.301C22, C4⋊C4.245C23, (C2×C8).199C23, (C2×C4).532C24, (C4×C8).277C22, C22⋊C4.116D4, C23.117(C2×D4), C4⋊Q8.164C22, C2.85(D4⋊6D4), C2.88(D4○SD16), (C4×D4).345C22, (C2×D4).252C23, (C2×Q8).237C23, (C4×Q8).174C22, C2.D8.196C22, C4.Q8.110C22, D4⋊C4.16C22, C23.20D4⋊10C2, C4⋊D4.101C22, C23.25D4⋊12C2, C23.19D4⋊10C2, C22⋊C8.210C22, (C22×C8).199C22, C22.792(C22×D4), C22⋊Q8.100C22, C22.50C24⋊7C2, (C22×C4).1164C23, Q8⋊C4.184C22, (C2×SD16).166C22, C42⋊C2.203C22, C22.49C24.4C2, C4⋊C4○(C4.Q8), C2.69(C2×C4○D8), C4.114(C2×C4○D4), (C2×C4).934(C2×D4), SmallGroup(128,2072)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.489C23
G = < a,b,c,d,e | a4=b4=1, c2=d2=a2, 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: 320 in 176 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C22×C8, C2×SD16, C23.25D4, C8×D4, C4×SD16, C8⋊8D4, D4⋊Q8, C4.Q16, C23.19D4, C23.20D4, C8⋊3Q8, C22.49C24, C22.50C24, C42.489C23
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, C42.489C23
►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 25 31 36)(2 26 32 33)(3 27 29 34)(4 28 30 35)(5 42 56 19)(6 43 53 20)(7 44 54 17)(8 41 55 18)(9 15 51 40)(10 16 52 37)(11 13 49 38)(12 14 50 39)(21 62 46 60)(22 63 47 57)(23 64 48 58)(24 61 45 59)
(1 57 3 59)(2 60 4 58)(5 52 7 50)(6 51 8 49)(9 55 11 53)(10 54 12 56)(13 43 15 41)(14 42 16 44)(17 39 19 37)(18 38 20 40)(21 35 23 33)(22 34 24 36)(25 47 27 45)(26 46 28 48)(29 61 31 63)(30 64 32 62)
(1 4 3 2)(5 18 7 20)(6 19 8 17)(9 10 11 12)(13 39 15 37)(14 40 16 38)(21 63 23 61)(22 64 24 62)(25 35 27 33)(26 36 28 34)(29 32 31 30)(41 54 43 56)(42 55 44 53)(45 60 47 58)(46 57 48 59)(49 50 51 52)
(1 10 31 52)(2 9 32 51)(3 12 29 50)(4 11 30 49)(5 57 56 63)(6 60 53 62)(7 59 54 61)(8 58 55 64)(13 35 38 28)(14 34 39 27)(15 33 40 26)(16 36 37 25)(17 45 44 24)(18 48 41 23)(19 47 42 22)(20 46 43 21)
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), (1,25,31,36)(2,26,32,33)(3,27,29,34)(4,28,30,35)(5,42,56,19)(6,43,53,20)(7,44,54,17)(8,41,55,18)(9,15,51,40)(10,16,52,37)(11,13,49,38)(12,14,50,39)(21,62,46,60)(22,63,47,57)(23,64,48,58)(24,61,45,59), (1,57,3,59)(2,60,4,58)(5,52,7,50)(6,51,8,49)(9,55,11,53)(10,54,12,56)(13,43,15,41)(14,42,16,44)(17,39,19,37)(18,38,20,40)(21,35,23,33)(22,34,24,36)(25,47,27,45)(26,46,28,48)(29,61,31,63)(30,64,32,62), (1,4,3,2)(5,18,7,20)(6,19,8,17)(9,10,11,12)(13,39,15,37)(14,40,16,38)(21,63,23,61)(22,64,24,62)(25,35,27,33)(26,36,28,34)(29,32,31,30)(41,54,43,56)(42,55,44,53)(45,60,47,58)(46,57,48,59)(49,50,51,52), (1,10,31,52)(2,9,32,51)(3,12,29,50)(4,11,30,49)(5,57,56,63)(6,60,53,62)(7,59,54,61)(8,58,55,64)(13,35,38,28)(14,34,39,27)(15,33,40,26)(16,36,37,25)(17,45,44,24)(18,48,41,23)(19,47,42,22)(20,46,43,21)>;
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), (1,25,31,36)(2,26,32,33)(3,27,29,34)(4,28,30,35)(5,42,56,19)(6,43,53,20)(7,44,54,17)(8,41,55,18)(9,15,51,40)(10,16,52,37)(11,13,49,38)(12,14,50,39)(21,62,46,60)(22,63,47,57)(23,64,48,58)(24,61,45,59), (1,57,3,59)(2,60,4,58)(5,52,7,50)(6,51,8,49)(9,55,11,53)(10,54,12,56)(13,43,15,41)(14,42,16,44)(17,39,19,37)(18,38,20,40)(21,35,23,33)(22,34,24,36)(25,47,27,45)(26,46,28,48)(29,61,31,63)(30,64,32,62), (1,4,3,2)(5,18,7,20)(6,19,8,17)(9,10,11,12)(13,39,15,37)(14,40,16,38)(21,63,23,61)(22,64,24,62)(25,35,27,33)(26,36,28,34)(29,32,31,30)(41,54,43,56)(42,55,44,53)(45,60,47,58)(46,57,48,59)(49,50,51,52), (1,10,31,52)(2,9,32,51)(3,12,29,50)(4,11,30,49)(5,57,56,63)(6,60,53,62)(7,59,54,61)(8,58,55,64)(13,35,38,28)(14,34,39,27)(15,33,40,26)(16,36,37,25)(17,45,44,24)(18,48,41,23)(19,47,42,22)(20,46,43,21) );
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)], [(1,25,31,36),(2,26,32,33),(3,27,29,34),(4,28,30,35),(5,42,56,19),(6,43,53,20),(7,44,54,17),(8,41,55,18),(9,15,51,40),(10,16,52,37),(11,13,49,38),(12,14,50,39),(21,62,46,60),(22,63,47,57),(23,64,48,58),(24,61,45,59)], [(1,57,3,59),(2,60,4,58),(5,52,7,50),(6,51,8,49),(9,55,11,53),(10,54,12,56),(13,43,15,41),(14,42,16,44),(17,39,19,37),(18,38,20,40),(21,35,23,33),(22,34,24,36),(25,47,27,45),(26,46,28,48),(29,61,31,63),(30,64,32,62)], [(1,4,3,2),(5,18,7,20),(6,19,8,17),(9,10,11,12),(13,39,15,37),(14,40,16,38),(21,63,23,61),(22,64,24,62),(25,35,27,33),(26,36,28,34),(29,32,31,30),(41,54,43,56),(42,55,44,53),(45,60,47,58),(46,57,48,59),(49,50,51,52)], [(1,10,31,52),(2,9,32,51),(3,12,29,50),(4,11,30,49),(5,57,56,63),(6,60,53,62),(7,59,54,61),(8,58,55,64),(13,35,38,28),(14,34,39,27),(15,33,40,26),(16,36,37,25),(17,45,44,24),(18,48,41,23),(19,47,42,22),(20,46,43,21)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | ··· | 4M | 4N | ··· | 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 | 4 | 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 | 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○SD16 |
| kernel | C42.489C23 | C23.25D4 | C8×D4 | C4×SD16 | C8⋊8D4 | D4⋊Q8 | C4.Q16 | C23.19D4 | C23.20D4 | C8⋊3Q8 | C22.49C24 | C22.50C24 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.489C23 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 11 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,11,6,0,0,0,0,0,4,0,0,4,0],[16,1,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,0,4,0,0,13,0] >;
C42.489C23 in GAP, Magma, Sage, TeX
C_4^2._{489}C_2^3 % in TeX
G:=Group("C4^2.489C2^3"); // GroupNames label
G:=SmallGroup(128,2072);
// by ID
G=gap.SmallGroup(128,2072);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,100,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=a^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