p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.67(C4×D4), D4⋊C4⋊7C4, C4⋊C4.212D4, C2.14(C4×SD16), C2.9(D8⋊C4), C2.3(C4⋊SD16), (C2×C4).112SD16, C23.781(C2×D4), C22.163(C4×D4), (C22×C4).696D4, C22.4Q16⋊17C2, C2.4(D4.2D4), C4.25(C4.4D4), C22.66(C4○D8), (C22×C8).49C22, C22.64(C2×SD16), C4.30(C42⋊2C2), C4.37(C42⋊C2), C22.85(C8⋊C22), (C2×C42).297C22, (C22×D4).37C22, C22.125(C4⋊D4), C22.7C42⋊29C2, (C22×C4).1380C23, C2.7(C23.19D4), C2.5(C23.46D4), C24.3C22.7C2, C2.11(C24.C22), C22.93(C22.D4), (C4×C4⋊C4)⋊7C2, (C2×C4.Q8)⋊18C2, C4⋊C4.150(C2×C4), (C2×C8).112(C2×C4), (C2×D4).95(C2×C4), (C2×D4⋊C4).6C2, (C2×C4).1010(C2×D4), (C2×C4).576(C4○D4), (C2×C4⋊C4).774C22, (C2×C4).398(C22×C4), SmallGroup(128,658)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.67(C4×D4)
G = < a,b,c,d | a4=b4=c4=1, d2=a, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a-1b, dcd-1=ac-1 >
Subgroups: 340 in 145 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, D4⋊C4, C4.Q8, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C22.4Q16, C4×C4⋊C4, C24.3C22, C2×D4⋊C4, C2×C4.Q8, C4.67(C4×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, SD16, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×SD16, C4○D8, C8⋊C22, C24.C22, C4×SD16, D8⋊C4, C4⋊SD16, D4.2D4, C23.46D4, C23.19D4, C4.67(C4×D4)
►On 64 points
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 26 51 18)(2 29 52 21)(3 32 53 24)(4 27 54 19)(5 30 55 22)(6 25 56 17)(7 28 49 20)(8 31 50 23)(9 47 57 33)(10 42 58 36)(11 45 59 39)(12 48 60 34)(13 43 61 37)(14 46 62 40)(15 41 63 35)(16 44 64 38)
(1 56 35 46)(2 45 36 55)(3 54 37 44)(4 43 38 53)(5 52 39 42)(6 41 40 51)(7 50 33 48)(8 47 34 49)(9 60 28 23)(10 22 29 59)(11 58 30 21)(12 20 31 57)(13 64 32 19)(14 18 25 63)(15 62 26 17)(16 24 27 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)
G:=sub<Sym(64)| (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,26,51,18)(2,29,52,21)(3,32,53,24)(4,27,54,19)(5,30,55,22)(6,25,56,17)(7,28,49,20)(8,31,50,23)(9,47,57,33)(10,42,58,36)(11,45,59,39)(12,48,60,34)(13,43,61,37)(14,46,62,40)(15,41,63,35)(16,44,64,38), (1,56,35,46)(2,45,36,55)(3,54,37,44)(4,43,38,53)(5,52,39,42)(6,41,40,51)(7,50,33,48)(8,47,34,49)(9,60,28,23)(10,22,29,59)(11,58,30,21)(12,20,31,57)(13,64,32,19)(14,18,25,63)(15,62,26,17)(16,24,27,61), (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)>;
G:=Group( (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,26,51,18)(2,29,52,21)(3,32,53,24)(4,27,54,19)(5,30,55,22)(6,25,56,17)(7,28,49,20)(8,31,50,23)(9,47,57,33)(10,42,58,36)(11,45,59,39)(12,48,60,34)(13,43,61,37)(14,46,62,40)(15,41,63,35)(16,44,64,38), (1,56,35,46)(2,45,36,55)(3,54,37,44)(4,43,38,53)(5,52,39,42)(6,41,40,51)(7,50,33,48)(8,47,34,49)(9,60,28,23)(10,22,29,59)(11,58,30,21)(12,20,31,57)(13,64,32,19)(14,18,25,63)(15,62,26,17)(16,24,27,61), (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) );
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,26,51,18),(2,29,52,21),(3,32,53,24),(4,27,54,19),(5,30,55,22),(6,25,56,17),(7,28,49,20),(8,31,50,23),(9,47,57,33),(10,42,58,36),(11,45,59,39),(12,48,60,34),(13,43,61,37),(14,46,62,40),(15,41,63,35),(16,44,64,38)], [(1,56,35,46),(2,45,36,55),(3,54,37,44),(4,43,38,53),(5,52,39,42),(6,41,40,51),(7,50,33,48),(8,47,34,49),(9,60,28,23),(10,22,29,59),(11,58,30,21),(12,20,31,57),(13,64,32,19),(14,18,25,63),(15,62,26,17),(16,24,27,61)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C4.67(C4×D4) | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C24.3C22 | C2×D4⋊C4 | C2×C4.Q8 | D4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 2 |
Matrix representation of C4.67(C4×D4) ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 15 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 15 | 0 | 0 |
| 0 | 0 | 16 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 5 | 5 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,12,12,0,0,0,0,12,5],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,15,4,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,16,0,0,0,0,15,4,0,0,0,0,0,0,5,5,0,0,0,0,12,5] >;
C4.67(C4×D4) in GAP, Magma, Sage, TeX
C_4._{67}(C_4\times D_4) % in TeX
G:=Group("C4.67(C4xD4)"); // GroupNames label
G:=SmallGroup(128,658);
// by ID
G=gap.SmallGroup(128,658);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,394,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a*c^-1>;
// generators/relations