p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.530C23, C4.1472+ 1+4, (C4×D8)⋊26C2, (C8×Q8)⋊18C2, C4⋊C4.288D4, (C4×Q16)⋊18C2, C4⋊SD16⋊46C2, C2.71(D4○D8), C8.96(C4○D4), C4.94(C4○D8), C8⋊4D4.11C2, (C2×Q8).191D4, D4.2D4⋊11C2, C8.12D4⋊11C2, C4⋊C4.447C23, C4⋊C8.332C22, (C2×C4).588C24, (C4×C8).128C22, (C2×C8).221C23, (C2×D8).42C22, C2.42(Q8⋊6D4), (C4×D4).221C22, (C2×D4).282C23, (C4×Q8).317C22, (C2×Q8).267C23, C2.D8.239C22, C4⋊1D4.107C22, (C2×Q16).145C22, C4.4D4.88C22, C22.848(C22×D4), D4⋊C4.192C22, C22.53C24⋊7C2, Q8⋊C4.168C22, (C2×SD16).104C22, C2.81(C2×C4○D8), C4.166(C2×C4○D4), (C2×C4).183(C2×D4), SmallGroup(128,2128)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.530C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2, e2=a2b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >
Subgroups: 408 in 192 conjugacy classes, 88 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C22.D4, C4.4D4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C2×Q16, C4×D8, C4×Q16, C8×Q8, C4⋊SD16, D4.2D4, C8⋊4D4, C8.12D4, C22.53C24, C42.530C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2+ 1+4, Q8⋊6D4, C2×C4○D8, D4○D8, C42.530C23
►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 53 47 58)(2 54 48 59)(3 55 45 60)(4 56 46 57)(5 42 61 52)(6 43 62 49)(7 44 63 50)(8 41 64 51)(9 34 21 37)(10 35 22 38)(11 36 23 39)(12 33 24 40)(13 28 18 31)(14 25 19 32)(15 26 20 29)(16 27 17 30)
(1 16 47 17)(2 20 48 15)(3 14 45 19)(4 18 46 13)(5 24 61 12)(6 11 62 23)(7 22 63 10)(8 9 64 21)(25 55 32 60)(26 59 29 54)(27 53 30 58)(28 57 31 56)(33 52 40 42)(34 41 37 51)(35 50 38 44)(36 43 39 49)
(1 50 3 52)(2 51 4 49)(5 58 7 60)(6 59 8 57)(9 18 11 20)(10 19 12 17)(13 23 15 21)(14 24 16 22)(25 33 27 35)(26 34 28 36)(29 37 31 39)(30 38 32 40)(41 46 43 48)(42 47 44 45)(53 63 55 61)(54 64 56 62)
(1 20 45 13)(2 14 46 17)(3 18 47 15)(4 16 48 19)(5 36 63 37)(6 38 64 33)(7 34 61 39)(8 40 62 35)(9 42 23 50)(10 51 24 43)(11 44 21 52)(12 49 22 41)(25 57 30 54)(26 55 31 58)(27 59 32 56)(28 53 29 60)
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), (1,53,47,58)(2,54,48,59)(3,55,45,60)(4,56,46,57)(5,42,61,52)(6,43,62,49)(7,44,63,50)(8,41,64,51)(9,34,21,37)(10,35,22,38)(11,36,23,39)(12,33,24,40)(13,28,18,31)(14,25,19,32)(15,26,20,29)(16,27,17,30), (1,16,47,17)(2,20,48,15)(3,14,45,19)(4,18,46,13)(5,24,61,12)(6,11,62,23)(7,22,63,10)(8,9,64,21)(25,55,32,60)(26,59,29,54)(27,53,30,58)(28,57,31,56)(33,52,40,42)(34,41,37,51)(35,50,38,44)(36,43,39,49), (1,50,3,52)(2,51,4,49)(5,58,7,60)(6,59,8,57)(9,18,11,20)(10,19,12,17)(13,23,15,21)(14,24,16,22)(25,33,27,35)(26,34,28,36)(29,37,31,39)(30,38,32,40)(41,46,43,48)(42,47,44,45)(53,63,55,61)(54,64,56,62), (1,20,45,13)(2,14,46,17)(3,18,47,15)(4,16,48,19)(5,36,63,37)(6,38,64,33)(7,34,61,39)(8,40,62,35)(9,42,23,50)(10,51,24,43)(11,44,21,52)(12,49,22,41)(25,57,30,54)(26,55,31,58)(27,59,32,56)(28,53,29,60)>;
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), (1,53,47,58)(2,54,48,59)(3,55,45,60)(4,56,46,57)(5,42,61,52)(6,43,62,49)(7,44,63,50)(8,41,64,51)(9,34,21,37)(10,35,22,38)(11,36,23,39)(12,33,24,40)(13,28,18,31)(14,25,19,32)(15,26,20,29)(16,27,17,30), (1,16,47,17)(2,20,48,15)(3,14,45,19)(4,18,46,13)(5,24,61,12)(6,11,62,23)(7,22,63,10)(8,9,64,21)(25,55,32,60)(26,59,29,54)(27,53,30,58)(28,57,31,56)(33,52,40,42)(34,41,37,51)(35,50,38,44)(36,43,39,49), (1,50,3,52)(2,51,4,49)(5,58,7,60)(6,59,8,57)(9,18,11,20)(10,19,12,17)(13,23,15,21)(14,24,16,22)(25,33,27,35)(26,34,28,36)(29,37,31,39)(30,38,32,40)(41,46,43,48)(42,47,44,45)(53,63,55,61)(54,64,56,62), (1,20,45,13)(2,14,46,17)(3,18,47,15)(4,16,48,19)(5,36,63,37)(6,38,64,33)(7,34,61,39)(8,40,62,35)(9,42,23,50)(10,51,24,43)(11,44,21,52)(12,49,22,41)(25,57,30,54)(26,55,31,58)(27,59,32,56)(28,53,29,60) );
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)], [(1,53,47,58),(2,54,48,59),(3,55,45,60),(4,56,46,57),(5,42,61,52),(6,43,62,49),(7,44,63,50),(8,41,64,51),(9,34,21,37),(10,35,22,38),(11,36,23,39),(12,33,24,40),(13,28,18,31),(14,25,19,32),(15,26,20,29),(16,27,17,30)], [(1,16,47,17),(2,20,48,15),(3,14,45,19),(4,18,46,13),(5,24,61,12),(6,11,62,23),(7,22,63,10),(8,9,64,21),(25,55,32,60),(26,59,29,54),(27,53,30,58),(28,57,31,56),(33,52,40,42),(34,41,37,51),(35,50,38,44),(36,43,39,49)], [(1,50,3,52),(2,51,4,49),(5,58,7,60),(6,59,8,57),(9,18,11,20),(10,19,12,17),(13,23,15,21),(14,24,16,22),(25,33,27,35),(26,34,28,36),(29,37,31,39),(30,38,32,40),(41,46,43,48),(42,47,44,45),(53,63,55,61),(54,64,56,62)], [(1,20,45,13),(2,14,46,17),(3,18,47,15),(4,16,48,19),(5,36,63,37),(6,38,64,33),(7,34,61,39),(8,40,62,35),(9,42,23,50),(10,51,24,43),(11,44,21,52),(12,49,22,41),(25,57,30,54),(26,55,31,58),(27,59,32,56),(28,53,29,60)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4O | 4P | 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 | 8 | 8 | 8 | 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | 2+ 1+4 | D4○D8 |
| kernel | C42.530C23 | C4×D8 | C4×Q16 | C8×Q8 | C4⋊SD16 | D4.2D4 | C8⋊4D4 | C8.12D4 | C22.53C24 | C4⋊C4 | C2×Q8 | C8 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 4 | 1 | 2 | 2 | 3 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.530C23 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 15 | 13 |
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 13 | 1 |
| 0 | 0 | 2 | 4 |
| 0 | 7 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 8 | 16 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,15,0,0,0,13],[16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[4,13,0,0,0,13,0,0,0,0,13,2,0,0,1,4],[0,12,0,0,7,0,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,4,16] >;
C42.530C23 in GAP, Magma, Sage, TeX
C_4^2._{530}C_2^3 % in TeX
G:=Group("C4^2.530C2^3"); // GroupNames label
G:=SmallGroup(128,2128);
// by ID
G=gap.SmallGroup(128,2128);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,436,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2,e^2=a^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,c*e=e*c,d*e=e*d>;
// generators/relations