p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.297D4, C4.52- 1+4, C42.431C23, C4.242+ 1+4, (C2×C4)⋊5Q16, C4.49(C2×Q16), C4.Q16⋊11C2, C4⋊2Q16⋊11C2, C22.4(C2×Q16), C4⋊C4.188C23, C4⋊C8.293C22, (C2×C8).172C23, (C2×C4).447C24, C8.18D4.7C2, (C22×C4).525D4, C23.406(C2×D4), C4⋊Q8.326C22, C2.17(C22×Q16), C2.D8.46C22, Q8⋊C4.9C22, (C2×Q8).177C23, (C2×Q16).30C22, (C4×Q8).124C22, (C22×C8).154C22, (C2×C42).904C22, C22.707(C22×D4), C22⋊Q8.214C22, C2.70(D8⋊C22), (C22×C4).1580C23, C23.37C23.43C2, C2.66(C22.31C24), (C2×C4⋊C8).43C2, (C2×C4).571(C2×D4), SmallGroup(128,1981)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.297D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2c3 >
Subgroups: 292 in 174 conjugacy classes, 94 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×Q16, C2×C4⋊C8, C4⋊2Q16, C8.18D4, C4.Q16, C23.37C23, C42.297D4
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C2×Q16, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C22×Q16, D8⋊C22, C42.297D4
►On 64 points
(1 7 5 3)(2 8 6 4)(9 40 13 36)(10 33 14 37)(11 34 15 38)(12 35 16 39)(17 23 21 19)(18 24 22 20)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 52 63 24)(2 21 64 49)(3 54 57 18)(4 23 58 51)(5 56 59 20)(6 17 60 53)(7 50 61 22)(8 19 62 55)(9 47 38 32)(10 29 39 44)(11 41 40 26)(12 31 33 46)(13 43 34 28)(14 25 35 48)(15 45 36 30)(16 27 37 42)
(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 31 5 27)(2 30 6 26)(3 29 7 25)(4 28 8 32)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 40 21 36)(18 39 22 35)(19 38 23 34)(20 37 24 33)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,23,21,19)(18,24,22,20)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,52,63,24)(2,21,64,49)(3,54,57,18)(4,23,58,51)(5,56,59,20)(6,17,60,53)(7,50,61,22)(8,19,62,55)(9,47,38,32)(10,29,39,44)(11,41,40,26)(12,31,33,46)(13,43,34,28)(14,25,35,48)(15,45,36,30)(16,27,37,42), (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), (1,31,5,27)(2,30,6,26)(3,29,7,25)(4,28,8,32)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,23,21,19)(18,24,22,20)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,52,63,24)(2,21,64,49)(3,54,57,18)(4,23,58,51)(5,56,59,20)(6,17,60,53)(7,50,61,22)(8,19,62,55)(9,47,38,32)(10,29,39,44)(11,41,40,26)(12,31,33,46)(13,43,34,28)(14,25,35,48)(15,45,36,30)(16,27,37,42), (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), (1,31,5,27)(2,30,6,26)(3,29,7,25)(4,28,8,32)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57) );
G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,40,13,36),(10,33,14,37),(11,34,15,38),(12,35,16,39),(17,23,21,19),(18,24,22,20),(25,42,29,46),(26,43,30,47),(27,44,31,48),(28,45,32,41),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,52,63,24),(2,21,64,49),(3,54,57,18),(4,23,58,51),(5,56,59,20),(6,17,60,53),(7,50,61,22),(8,19,62,55),(9,47,38,32),(10,29,39,44),(11,41,40,26),(12,31,33,46),(13,43,34,28),(14,25,35,48),(15,45,36,30),(16,27,37,42)], [(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)], [(1,31,5,27),(2,30,6,26),(3,29,7,25),(4,28,8,32),(9,51,13,55),(10,50,14,54),(11,49,15,53),(12,56,16,52),(17,40,21,36),(18,39,22,35),(19,38,23,34),(20,37,24,33),(41,64,45,60),(42,63,46,59),(43,62,47,58),(44,61,48,57)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q16 | 2+ 1+4 | 2- 1+4 | D8⋊C22 |
| kernel | C42.297D4 | C2×C4⋊C8 | C4⋊2Q16 | C8.18D4 | C4.Q16 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C42.297D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 8 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 |
| 0 | 0 | 0 | 0 | 9 | 4 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 3 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,4,8,0,0,0,0,13,13,0,0,0,0,0,0,13,9,0,0,0,0,4,4],[15,5,0,0,0,0,0,8,0,0,0,0,0,0,8,16,0,0,0,0,0,9,0,0,0,0,0,0,15,13,0,0,0,0,0,2],[9,3,0,0,0,0,1,8,0,0,0,0,0,0,0,0,8,16,0,0,0,0,0,9,0,0,2,4,0,0,0,0,0,15,0,0] >;
C42.297D4 in GAP, Magma, Sage, TeX
C_4^2._{297}D_4 % in TeX
G:=Group("C4^2.297D4"); // GroupNames label
G:=SmallGroup(128,1981);
// by ID
G=gap.SmallGroup(128,1981);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,352,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^3>;
// generators/relations