p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.212D4, C42.322C23, (C2×Q8)⋊28D4, Q8.4(C2×D4), C4⋊SD16⋊1C2, C4⋊2Q16⋊18C2, C4⋊C8.38C22, C4⋊4(C8.C22), (C2×C8).12C23, C4.68(C22×D4), C4⋊C4.378C23, C4⋊M4(2)⋊3C2, (C2×C4).241C24, Q8.D4⋊12C2, (C2×D4).50C23, (C22×C4).794D4, C23.653(C2×D4), C4⋊Q8.256C22, C4.169(C4⋊D4), C23.36D4⋊6C2, (C2×Q16).51C22, (C2×Q8).358C23, (C4×Q8).291C22, (C2×SD16).3C22, D4⋊C4.20C22, C4⋊1D4.137C22, C22.33(C4⋊D4), (C22×C4).971C23, (C2×C42).810C22, Q8⋊C4.22C22, C22.501(C22×D4), C2.12(D8⋊C22), C4.4D4.126C22, (C2×M4(2)).48C22, (C22×Q8).470C22, C22.26C24.30C2, (C2×C4×Q8)⋊36C2, C4.151(C2×C4○D4), C2.59(C2×C4⋊D4), (C2×C4).1420(C2×D4), (C2×C8.C22)⋊14C2, C2.15(C2×C8.C22), (C2×C4).272(C4○D4), (C2×C4⋊C4).921C22, (C2×C4○D4).116C22, SmallGroup(128,1769)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 444 in 242 conjugacy classes, 102 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×19], D4 [×12], Q8 [×4], Q8 [×10], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×8], C2×Q8 [×3], C4○D4 [×8], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C2×SD16 [×4], C2×Q16 [×4], C8.C22 [×8], C22×Q8, C2×C4○D4 [×2], C23.36D4 [×2], C4⋊M4(2), C4⋊SD16 [×2], C4⋊2Q16 [×2], Q8.D4 [×4], C2×C4×Q8, C22.26C24, C2×C8.C22 [×2], C42.212D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8.C22 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8.C22, D8⋊C22, C42.212D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, cbc-1=b-1, bd=db, dcd=c3 >
►On 64 points
(1 54 29 61)(2 62 30 55)(3 56 31 63)(4 64 32 49)(5 50 25 57)(6 58 26 51)(7 52 27 59)(8 60 28 53)(9 20 36 48)(10 41 37 21)(11 22 38 42)(12 43 39 23)(13 24 40 44)(14 45 33 17)(15 18 34 46)(16 47 35 19)
(1 10 5 14)(2 15 6 11)(3 12 7 16)(4 9 8 13)(17 61 21 57)(18 58 22 62)(19 63 23 59)(20 60 24 64)(25 33 29 37)(26 38 30 34)(27 35 31 39)(28 40 32 36)(41 50 45 54)(42 55 46 51)(43 52 47 56)(44 49 48 53)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 45)(18 48)(19 43)(20 46)(21 41)(22 44)(23 47)(24 42)(26 28)(27 31)(30 32)(34 36)(35 39)(38 40)(49 62)(50 57)(51 60)(52 63)(53 58)(54 61)(55 64)(56 59)
G:=sub<Sym(64)| (1,54,29,61)(2,62,30,55)(3,56,31,63)(4,64,32,49)(5,50,25,57)(6,58,26,51)(7,52,27,59)(8,60,28,53)(9,20,36,48)(10,41,37,21)(11,22,38,42)(12,43,39,23)(13,24,40,44)(14,45,33,17)(15,18,34,46)(16,47,35,19), (1,10,5,14)(2,15,6,11)(3,12,7,16)(4,9,8,13)(17,61,21,57)(18,58,22,62)(19,63,23,59)(20,60,24,64)(25,33,29,37)(26,38,30,34)(27,35,31,39)(28,40,32,36)(41,50,45,54)(42,55,46,51)(43,52,47,56)(44,49,48,53), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,45)(18,48)(19,43)(20,46)(21,41)(22,44)(23,47)(24,42)(26,28)(27,31)(30,32)(34,36)(35,39)(38,40)(49,62)(50,57)(51,60)(52,63)(53,58)(54,61)(55,64)(56,59)>;
G:=Group( (1,54,29,61)(2,62,30,55)(3,56,31,63)(4,64,32,49)(5,50,25,57)(6,58,26,51)(7,52,27,59)(8,60,28,53)(9,20,36,48)(10,41,37,21)(11,22,38,42)(12,43,39,23)(13,24,40,44)(14,45,33,17)(15,18,34,46)(16,47,35,19), (1,10,5,14)(2,15,6,11)(3,12,7,16)(4,9,8,13)(17,61,21,57)(18,58,22,62)(19,63,23,59)(20,60,24,64)(25,33,29,37)(26,38,30,34)(27,35,31,39)(28,40,32,36)(41,50,45,54)(42,55,46,51)(43,52,47,56)(44,49,48,53), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,45)(18,48)(19,43)(20,46)(21,41)(22,44)(23,47)(24,42)(26,28)(27,31)(30,32)(34,36)(35,39)(38,40)(49,62)(50,57)(51,60)(52,63)(53,58)(54,61)(55,64)(56,59) );
G=PermutationGroup([(1,54,29,61),(2,62,30,55),(3,56,31,63),(4,64,32,49),(5,50,25,57),(6,58,26,51),(7,52,27,59),(8,60,28,53),(9,20,36,48),(10,41,37,21),(11,22,38,42),(12,43,39,23),(13,24,40,44),(14,45,33,17),(15,18,34,46),(16,47,35,19)], [(1,10,5,14),(2,15,6,11),(3,12,7,16),(4,9,8,13),(17,61,21,57),(18,58,22,62),(19,63,23,59),(20,60,24,64),(25,33,29,37),(26,38,30,34),(27,35,31,39),(28,40,32,36),(41,50,45,54),(42,55,46,51),(43,52,47,56),(44,49,48,53)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,45),(18,48),(19,43),(20,46),(21,41),(22,44),(23,47),(24,42),(26,28),(27,31),(30,32),(34,36),(35,39),(38,40),(49,62),(50,57),(51,60),(52,63),(53,58),(54,61),(55,64),(56,59)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 4 | 13 | 9 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 10 |
| 0 | 0 | 12 | 12 | 10 | 10 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 12 | 7 | 5 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 1 | 1 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,4,4,0,0,0,0,4,0,0,0,0,4,13,0,0,0,0,0,9,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,16,16,0,16,0,0,15,0,0,16],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,12,12,5,12,0,0,12,12,5,7,0,0,0,10,0,5,0,0,10,10,0,10],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,1,0,0,0,0,0,1] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 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 | D4 | C4○D4 | C8.C22 | D8⋊C22 |
| kernel | C42.212D4 | C23.36D4 | C4⋊M4(2) | C4⋊SD16 | C4⋊2Q16 | Q8.D4 | C2×C4×Q8 | C22.26C24 | C2×C8.C22 | C42 | C22×C4 | C2×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{212}D_4 % in TeX
G:=Group("C4^2.212D4"); // GroupNames label
G:=SmallGroup(128,1769);
// by ID
G=gap.SmallGroup(128,1769);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,2019,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^3>;
// generators/relations