p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.305D4, C42.615C23, D4⋊C8⋊31C2, Q8⋊C8⋊35C2, C22.3C4≀C2, C4.3(C8○D4), C4⋊D4.6C4, C22⋊Q8.6C4, C4.4D4.5C4, (C4×D4).6C22, C42.C2.7C4, (C4×Q8).6C22, C4.121(C4○D8), C4⋊C8.199C22, C42.259(C2×C4), (C4×C8).365C22, (C22×C4).540D4, C42.6C4⋊27C2, C23.99(C22⋊C4), (C2×C42).1037C22, C2.7(C23.24D4), C23.36C23.3C2, (C2×C4×C8)⋊4C2, C2.14(C2×C4≀C2), C4⋊C4.55(C2×C4), (C2×D4).53(C2×C4), (C2×Q8).48(C2×C4), (C2×C4).1143(C2×D4), (C2×C4).320(C22×C4), (C22×C4).398(C2×C4), (C2×C4).169(C22⋊C4), C22.170(C2×C22⋊C4), C2.20((C22×C8)⋊C2), SmallGroup(128,226)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.305D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 212 in 115 conjugacy classes, 48 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, D4⋊C8, Q8⋊C8, C2×C4×C8, C42.6C4, C23.36C23, C42.305D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C8○D4, C4○D8, (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C42.305D4
►On 64 points
(1 43 17 35)(2 44 18 36)(3 45 19 37)(4 46 20 38)(5 47 21 39)(6 48 22 40)(7 41 23 33)(8 42 24 34)(9 28 50 63)(10 29 51 64)(11 30 52 57)(12 31 53 58)(13 32 54 59)(14 25 55 60)(15 26 56 61)(16 27 49 62)
(1 30 21 61)(2 31 22 62)(3 32 23 63)(4 25 24 64)(5 26 17 57)(6 27 18 58)(7 28 19 59)(8 29 20 60)(9 45 54 33)(10 46 55 34)(11 47 56 35)(12 48 49 36)(13 41 50 37)(14 42 51 38)(15 43 52 39)(16 44 53 40)
(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 29 30 20 21 60 61 8)(2 19 31 59 22 7 62 28)(3 58 32 6 23 27 63 18)(4 5 25 26 24 17 64 57)(9 48 45 49 54 36 33 12)(10 56 46 35 55 11 34 47)(13 44 41 53 50 40 37 16)(14 52 42 39 51 15 38 43)
G:=sub<Sym(64)| (1,43,17,35)(2,44,18,36)(3,45,19,37)(4,46,20,38)(5,47,21,39)(6,48,22,40)(7,41,23,33)(8,42,24,34)(9,28,50,63)(10,29,51,64)(11,30,52,57)(12,31,53,58)(13,32,54,59)(14,25,55,60)(15,26,56,61)(16,27,49,62), (1,30,21,61)(2,31,22,62)(3,32,23,63)(4,25,24,64)(5,26,17,57)(6,27,18,58)(7,28,19,59)(8,29,20,60)(9,45,54,33)(10,46,55,34)(11,47,56,35)(12,48,49,36)(13,41,50,37)(14,42,51,38)(15,43,52,39)(16,44,53,40), (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), (1,29,30,20,21,60,61,8)(2,19,31,59,22,7,62,28)(3,58,32,6,23,27,63,18)(4,5,25,26,24,17,64,57)(9,48,45,49,54,36,33,12)(10,56,46,35,55,11,34,47)(13,44,41,53,50,40,37,16)(14,52,42,39,51,15,38,43)>;
G:=Group( (1,43,17,35)(2,44,18,36)(3,45,19,37)(4,46,20,38)(5,47,21,39)(6,48,22,40)(7,41,23,33)(8,42,24,34)(9,28,50,63)(10,29,51,64)(11,30,52,57)(12,31,53,58)(13,32,54,59)(14,25,55,60)(15,26,56,61)(16,27,49,62), (1,30,21,61)(2,31,22,62)(3,32,23,63)(4,25,24,64)(5,26,17,57)(6,27,18,58)(7,28,19,59)(8,29,20,60)(9,45,54,33)(10,46,55,34)(11,47,56,35)(12,48,49,36)(13,41,50,37)(14,42,51,38)(15,43,52,39)(16,44,53,40), (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), (1,29,30,20,21,60,61,8)(2,19,31,59,22,7,62,28)(3,58,32,6,23,27,63,18)(4,5,25,26,24,17,64,57)(9,48,45,49,54,36,33,12)(10,56,46,35,55,11,34,47)(13,44,41,53,50,40,37,16)(14,52,42,39,51,15,38,43) );
G=PermutationGroup([[(1,43,17,35),(2,44,18,36),(3,45,19,37),(4,46,20,38),(5,47,21,39),(6,48,22,40),(7,41,23,33),(8,42,24,34),(9,28,50,63),(10,29,51,64),(11,30,52,57),(12,31,53,58),(13,32,54,59),(14,25,55,60),(15,26,56,61),(16,27,49,62)], [(1,30,21,61),(2,31,22,62),(3,32,23,63),(4,25,24,64),(5,26,17,57),(6,27,18,58),(7,28,19,59),(8,29,20,60),(9,45,54,33),(10,46,55,34),(11,47,56,35),(12,48,49,36),(13,41,50,37),(14,42,51,38),(15,43,52,39),(16,44,53,40)], [(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)], [(1,29,30,20,21,60,61,8),(2,19,31,59,22,7,62,28),(3,58,32,6,23,27,63,18),(4,5,25,26,24,17,64,57),(9,48,45,49,54,36,33,12),(10,56,46,35,55,11,34,47),(13,44,41,53,50,40,37,16),(14,52,42,39,51,15,38,43)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | C8○D4 | C4○D8 | C4≀C2 |
| kernel | C42.305D4 | D4⋊C8 | Q8⋊C8 | C2×C4×C8 | C42.6C4 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C4.4D4 | C42.C2 | C42 | C22×C4 | C4 | C4 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of C42.305D4 ►in GL4(𝔽17) generated by
| 16 | 15 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 6 | 11 |
| 2 | 0 | 0 | 0 |
| 15 | 15 | 0 | 0 |
| 0 | 0 | 11 | 3 |
| 0 | 0 | 11 | 6 |
G:=sub<GL(4,GF(17))| [16,0,0,0,15,1,0,0,0,0,13,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,0,6,0,0,14,11],[2,15,0,0,0,15,0,0,0,0,11,11,0,0,3,6] >;
C42.305D4 in GAP, Magma, Sage, TeX
C_4^2._{305}D_4 % in TeX
G:=Group("C4^2.305D4"); // GroupNames label
G:=SmallGroup(128,226);
// by ID
G=gap.SmallGroup(128,226);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,723,184,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations