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, C23.653(C2×D4), (C22×C4).794D4, C4⋊Q8.256C22, C4.169(C4⋊D4), C23.36D4⋊6C2, (C4×Q8).291C22, (C2×Q16).51C22, (C2×Q8).358C23, (C2×SD16).3C22, D4⋊C4.20C22, C4⋊1D4.137C22, C22.33(C4⋊D4), (C2×C42).810C22, (C22×C4).971C23, 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
Generators and relations for C42.212D4
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 >
Subgroups: 444 in 242 conjugacy classes, 102 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C23.36D4, C4⋊M4(2), C4⋊SD16, C4⋊2Q16, Q8.D4, C2×C4×Q8, C22.26C24, C2×C8.C22, C42.212D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C8.C22, C22×D4, C2×C4○D4, C2×C4⋊D4, C2×C8.C22, D8⋊C22, C42.212D4
►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 41 37 21)(10 22 38 42)(11 43 39 23)(12 24 40 44)(13 45 33 17)(14 18 34 46)(15 47 35 19)(16 20 36 48)
(1 9 5 13)(2 14 6 10)(3 11 7 15)(4 16 8 12)(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)(10 12)(11 15)(14 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)(9,41,37,21)(10,22,38,42)(11,43,39,23)(12,24,40,44)(13,45,33,17)(14,18,34,46)(15,47,35,19)(16,20,36,48), (1,9,5,13)(2,14,6,10)(3,11,7,15)(4,16,8,12)(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)(10,12)(11,15)(14,16)(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)(9,41,37,21)(10,22,38,42)(11,43,39,23)(12,24,40,44)(13,45,33,17)(14,18,34,46)(15,47,35,19)(16,20,36,48), (1,9,5,13)(2,14,6,10)(3,11,7,15)(4,16,8,12)(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)(10,12)(11,15)(14,16)(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),(9,41,37,21),(10,22,38,42),(11,43,39,23),(12,24,40,44),(13,45,33,17),(14,18,34,46),(15,47,35,19),(16,20,36,48)], [(1,9,5,13),(2,14,6,10),(3,11,7,15),(4,16,8,12),(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),(10,12),(11,15),(14,16),(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)]])
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 |
Matrix representation of C42.212D4 ►in 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] >;
C42.212D4 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