p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.373D4, C42.603C23, D4⋊C8⋊4C2, Q8⋊C8⋊4C2, C4⋊D4.2C4, (C4×C8).5C22, C22⋊Q8.2C4, C4.26(C8○D4), C42.56(C2×C4), C4.4D4.3C4, (C4×D4).2C22, C42.C2.5C4, (C4×Q8).2C22, C4.119(C4○D8), C4⋊C8.249C22, (C22×C4).201D4, C42.12C4⋊9C2, C23.43(C22⋊C4), (C2×C42).159C22, C2.8(C42⋊C22), C2.5(C23.24D4), C23.36C23.1C2, C4⋊C4.49(C2×C4), (C2×D4).49(C2×C4), (C2×Q8).44(C2×C4), (C2×C4).1446(C2×D4), (C2×C4).76(C22⋊C4), (C22×C4).181(C2×C4), (C2×C4).308(C22×C4), C22.158(C2×C22⋊C4), C2.14((C22×C8)⋊C2), SmallGroup(128,214)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.373D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 204 in 106 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2, C4, C4, 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, C22⋊C8, C4⋊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, D4⋊C8, Q8⋊C8, C42.12C4, C23.36C23, C42.373D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8○D4, C4○D8, (C22×C8)⋊C2, C23.24D4, C42⋊C22, C42.373D4
►On 64 points
(1 43 51 35)(2 40 52 48)(3 45 53 37)(4 34 54 42)(5 47 55 39)(6 36 56 44)(7 41 49 33)(8 38 50 46)(9 20 32 63)(10 60 25 17)(11 22 26 57)(12 62 27 19)(13 24 28 59)(14 64 29 21)(15 18 30 61)(16 58 31 23)
(1 22 55 61)(2 23 56 62)(3 24 49 63)(4 17 50 64)(5 18 51 57)(6 19 52 58)(7 20 53 59)(8 21 54 60)(9 45 28 33)(10 46 29 34)(11 47 30 35)(12 48 31 36)(13 41 32 37)(14 42 25 38)(15 43 26 39)(16 44 27 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 21 22 54 55 60 61 8)(2 53 23 59 56 7 62 20)(3 58 24 6 49 19 63 52)(4 5 17 18 50 51 64 57)(9 48 45 31 28 36 33 12)(10 30 46 35 29 11 34 47)(13 44 41 27 32 40 37 16)(14 26 42 39 25 15 38 43)
G:=sub<Sym(64)| (1,43,51,35)(2,40,52,48)(3,45,53,37)(4,34,54,42)(5,47,55,39)(6,36,56,44)(7,41,49,33)(8,38,50,46)(9,20,32,63)(10,60,25,17)(11,22,26,57)(12,62,27,19)(13,24,28,59)(14,64,29,21)(15,18,30,61)(16,58,31,23), (1,22,55,61)(2,23,56,62)(3,24,49,63)(4,17,50,64)(5,18,51,57)(6,19,52,58)(7,20,53,59)(8,21,54,60)(9,45,28,33)(10,46,29,34)(11,47,30,35)(12,48,31,36)(13,41,32,37)(14,42,25,38)(15,43,26,39)(16,44,27,40), (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), (1,21,22,54,55,60,61,8)(2,53,23,59,56,7,62,20)(3,58,24,6,49,19,63,52)(4,5,17,18,50,51,64,57)(9,48,45,31,28,36,33,12)(10,30,46,35,29,11,34,47)(13,44,41,27,32,40,37,16)(14,26,42,39,25,15,38,43)>;
G:=Group( (1,43,51,35)(2,40,52,48)(3,45,53,37)(4,34,54,42)(5,47,55,39)(6,36,56,44)(7,41,49,33)(8,38,50,46)(9,20,32,63)(10,60,25,17)(11,22,26,57)(12,62,27,19)(13,24,28,59)(14,64,29,21)(15,18,30,61)(16,58,31,23), (1,22,55,61)(2,23,56,62)(3,24,49,63)(4,17,50,64)(5,18,51,57)(6,19,52,58)(7,20,53,59)(8,21,54,60)(9,45,28,33)(10,46,29,34)(11,47,30,35)(12,48,31,36)(13,41,32,37)(14,42,25,38)(15,43,26,39)(16,44,27,40), (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), (1,21,22,54,55,60,61,8)(2,53,23,59,56,7,62,20)(3,58,24,6,49,19,63,52)(4,5,17,18,50,51,64,57)(9,48,45,31,28,36,33,12)(10,30,46,35,29,11,34,47)(13,44,41,27,32,40,37,16)(14,26,42,39,25,15,38,43) );
G=PermutationGroup([[(1,43,51,35),(2,40,52,48),(3,45,53,37),(4,34,54,42),(5,47,55,39),(6,36,56,44),(7,41,49,33),(8,38,50,46),(9,20,32,63),(10,60,25,17),(11,22,26,57),(12,62,27,19),(13,24,28,59),(14,64,29,21),(15,18,30,61),(16,58,31,23)], [(1,22,55,61),(2,23,56,62),(3,24,49,63),(4,17,50,64),(5,18,51,57),(6,19,52,58),(7,20,53,59),(8,21,54,60),(9,45,28,33),(10,46,29,34),(11,47,30,35),(12,48,31,36),(13,41,32,37),(14,42,25,38),(15,43,26,39),(16,44,27,40)], [(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)], [(1,21,22,54,55,60,61,8),(2,53,23,59,56,7,62,20),(3,58,24,6,49,19,63,52),(4,5,17,18,50,51,64,57),(9,48,45,31,28,36,33,12),(10,30,46,35,29,11,34,47),(13,44,41,27,32,40,37,16),(14,26,42,39,25,15,38,43)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | C8○D4 | C4○D8 | C42⋊C22 |
| kernel | C42.373D4 | D4⋊C8 | Q8⋊C8 | C42.12C4 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C4.4D4 | C42.C2 | C42 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.373D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 8 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[3,3,0,0,14,3,0,0,0,0,9,0,0,0,0,8],[3,14,0,0,14,14,0,0,0,0,9,0,0,0,0,9] >;
C42.373D4 in GAP, Magma, Sage, TeX
C_4^2._{373}D_4 % in TeX
G:=Group("C4^2.373D4"); // GroupNames label
G:=SmallGroup(128,214);
// by ID
G=gap.SmallGroup(128,214);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,184,1123,570,136,172]);
// 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,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations