p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.445D4, C42.324C23, D4.5(C2×D4), Q8.5(C2×D4), C4○D4.29D4, C4⋊2Q16⋊19C2, D4.D4⋊2C2, C4⋊C8.39C22, C4⋊5(C8.C22), (C2×C8).14C23, C4.70(C22×D4), C4⋊C4.380C23, C4⋊M4(2)⋊5C2, (C2×C4).243C24, (C22×C4).423D4, C23.655(C2×D4), C4⋊Q8.257C22, (C2×Q8).37C23, C4.105(C4⋊D4), (C2×D4).386C23, (C4×D4).311C22, C23.38D4⋊6C2, (C2×Q16).52C22, (C4×Q8).292C22, (C2×SD16).4C22, C22.78(C4⋊D4), (C22×C4).973C23, (C2×C42).812C22, Q8⋊C4.23C22, C22.503(C22×D4), (C2×M4(2)).50C22, (C22×Q8).271C22, C42⋊C2.312C22, (C2×C4⋊Q8)⋊30C2, (C4×C4○D4).22C2, C4.153(C2×C4○D4), C2.61(C2×C4⋊D4), (C2×C4).1422(C2×D4), C2.16(C2×C8.C22), (C2×C4).274(C4○D4), (C2×C8.C22).10C2, (C2×C4○D4).299C22, SmallGroup(128,1771)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 420 in 240 conjugacy classes, 104 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×23], D4 [×2], D4 [×5], Q8 [×2], Q8 [×13], C23, C23, C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×10], C2×C8 [×4], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×4], C4○D4 [×2], Q8⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×4], C2×Q16 [×4], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4, C23.38D4 [×2], C4⋊M4(2), D4.D4 [×4], C4⋊2Q16 [×4], C4×C4○D4, C2×C4⋊Q8, C2×C8.C22 [×2], C42.445D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8.C22 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8.C22 [×2], C42.445D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=c3 >
►On 64 points
(1 59 51 22)(2 23 52 60)(3 61 53 24)(4 17 54 62)(5 63 55 18)(6 19 56 64)(7 57 49 20)(8 21 50 58)(9 29 33 42)(10 43 34 30)(11 31 35 44)(12 45 36 32)(13 25 37 46)(14 47 38 26)(15 27 39 48)(16 41 40 28)
(1 43 5 47)(2 48 6 44)(3 45 7 41)(4 42 8 46)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)(25 54 29 50)(26 51 30 55)(27 56 31 52)(28 53 32 49)(33 58 37 62)(34 63 38 59)(35 60 39 64)(36 57 40 61)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 18 21 22)(19 24 23 20)(25 26 29 30)(27 32 31 28)(33 34 37 38)(35 40 39 36)(41 48 45 44)(42 43 46 47)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)
G:=sub<Sym(64)| (1,59,51,22)(2,23,52,60)(3,61,53,24)(4,17,54,62)(5,63,55,18)(6,19,56,64)(7,57,49,20)(8,21,50,58)(9,29,33,42)(10,43,34,30)(11,31,35,44)(12,45,36,32)(13,25,37,46)(14,47,38,26)(15,27,39,48)(16,41,40,28), (1,43,5,47)(2,48,6,44)(3,45,7,41)(4,42,8,46)(9,21,13,17)(10,18,14,22)(11,23,15,19)(12,20,16,24)(25,54,29,50)(26,51,30,55)(27,56,31,52)(28,53,32,49)(33,58,37,62)(34,63,38,59)(35,60,39,64)(36,57,40,61), (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), (1,4,5,8)(2,7,6,3)(9,10,13,14)(11,16,15,12)(17,18,21,22)(19,24,23,20)(25,26,29,30)(27,32,31,28)(33,34,37,38)(35,40,39,36)(41,48,45,44)(42,43,46,47)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63)>;
G:=Group( (1,59,51,22)(2,23,52,60)(3,61,53,24)(4,17,54,62)(5,63,55,18)(6,19,56,64)(7,57,49,20)(8,21,50,58)(9,29,33,42)(10,43,34,30)(11,31,35,44)(12,45,36,32)(13,25,37,46)(14,47,38,26)(15,27,39,48)(16,41,40,28), (1,43,5,47)(2,48,6,44)(3,45,7,41)(4,42,8,46)(9,21,13,17)(10,18,14,22)(11,23,15,19)(12,20,16,24)(25,54,29,50)(26,51,30,55)(27,56,31,52)(28,53,32,49)(33,58,37,62)(34,63,38,59)(35,60,39,64)(36,57,40,61), (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), (1,4,5,8)(2,7,6,3)(9,10,13,14)(11,16,15,12)(17,18,21,22)(19,24,23,20)(25,26,29,30)(27,32,31,28)(33,34,37,38)(35,40,39,36)(41,48,45,44)(42,43,46,47)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63) );
G=PermutationGroup([(1,59,51,22),(2,23,52,60),(3,61,53,24),(4,17,54,62),(5,63,55,18),(6,19,56,64),(7,57,49,20),(8,21,50,58),(9,29,33,42),(10,43,34,30),(11,31,35,44),(12,45,36,32),(13,25,37,46),(14,47,38,26),(15,27,39,48),(16,41,40,28)], [(1,43,5,47),(2,48,6,44),(3,45,7,41),(4,42,8,46),(9,21,13,17),(10,18,14,22),(11,23,15,19),(12,20,16,24),(25,54,29,50),(26,51,30,55),(27,56,31,52),(28,53,32,49),(33,58,37,62),(34,63,38,59),(35,60,39,64),(36,57,40,61)], [(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)], [(1,4,5,8),(2,7,6,3),(9,10,13,14),(11,16,15,12),(17,18,21,22),(19,24,23,20),(25,26,29,30),(27,32,31,28),(33,34,37,38),(35,40,39,36),(41,48,45,44),(42,43,46,47),(49,56,53,52),(50,51,54,55),(57,64,61,60),(58,59,62,63)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 1 | 16 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 6 |
| 0 | 0 | 16 | 10 | 6 | 11 |
| 0 | 0 | 16 | 1 | 0 | 8 |
| 0 | 0 | 10 | 8 | 10 | 14 |
| 5 | 16 | 0 | 0 | 0 | 0 |
| 7 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 6 |
| 0 | 0 | 1 | 7 | 11 | 6 |
| 0 | 0 | 16 | 1 | 0 | 8 |
| 0 | 0 | 16 | 0 | 7 | 0 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,16,1,1,0,0,0,1,0,16,0,0,16,16,0,0,0,0,0,2,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[12,10,0,0,0,0,1,5,0,0,0,0,0,0,10,16,16,10,0,0,7,10,1,8,0,0,0,6,0,10,0,0,6,11,8,14],[5,7,0,0,0,0,16,12,0,0,0,0,0,0,10,1,16,16,0,0,7,7,1,0,0,0,0,11,0,7,0,0,6,6,8,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C8.C22 |
| kernel | C42.445D4 | C23.38D4 | C4⋊M4(2) | D4.D4 | C4⋊2Q16 | C4×C4○D4 | C2×C4⋊Q8 | C2×C8.C22 | C42 | C22×C4 | C4○D4 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{445}D_4 % in TeX
G:=Group("C4^2.445D4"); // GroupNames label
G:=SmallGroup(128,1771);
// by ID
G=gap.SmallGroup(128,1771);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,2019,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations