p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.370D4, C4⋊C8.3C4, C4.39C4≀C2, C8⋊C4.10C4, C22⋊C8.10C4, (C2×C4).14C42, C42.40(C2×C4), (C22×C4).25Q8, C23.45(C4⋊C4), (C22×C4).643D4, C2.13(C42⋊6C4), C42.6C4.10C2, C22.7(C8.C4), C2.9(C4.C42), (C2×C42).137C22, C42.12C4.11C2, C2.10(M4(2)⋊4C4), C22.52(C2.C42), (C2×C4).22(C4⋊C4), (C2×C8⋊C4).16C2, (C22×C4).157(C2×C4), (C2×C4).375(C22⋊C4), SmallGroup(128,34)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.370D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=ba=ab, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=ab2c3 >
Subgroups: 120 in 72 conjugacy classes, 34 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C8⋊C4, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C2×C8⋊C4, C42.12C4, C42.6C4, C42.370D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C4≀C2, C8.C4, C42⋊6C4, C4.C42, M4(2)⋊4C4, C42.370D4
►On 64 points
(1 47 27 12)(2 44 28 9)(3 41 29 14)(4 46 30 11)(5 43 31 16)(6 48 32 13)(7 45 25 10)(8 42 26 15)(17 36 54 57)(18 33 55 62)(19 38 56 59)(20 35 49 64)(21 40 50 61)(22 37 51 58)(23 34 52 63)(24 39 53 60)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(25 31 29 27)(26 32 30 28)(33 60 37 64)(34 61 38 57)(35 62 39 58)(36 63 40 59)(41 47 45 43)(42 48 46 44)
(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 58 45 49 31 33 14 24)(2 17 42 63 32 50 11 38)(3 39 47 22 25 64 16 55)(4 56 44 36 26 23 13 61)(5 62 41 53 27 37 10 20)(6 21 46 59 28 54 15 34)(7 35 43 18 29 60 12 51)(8 52 48 40 30 19 9 57)
G:=sub<Sym(64)| (1,47,27,12)(2,44,28,9)(3,41,29,14)(4,46,30,11)(5,43,31,16)(6,48,32,13)(7,45,25,10)(8,42,26,15)(17,36,54,57)(18,33,55,62)(19,38,56,59)(20,35,49,64)(21,40,50,61)(22,37,51,58)(23,34,52,63)(24,39,53,60), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(25,31,29,27)(26,32,30,28)(33,60,37,64)(34,61,38,57)(35,62,39,58)(36,63,40,59)(41,47,45,43)(42,48,46,44), (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), (1,58,45,49,31,33,14,24)(2,17,42,63,32,50,11,38)(3,39,47,22,25,64,16,55)(4,56,44,36,26,23,13,61)(5,62,41,53,27,37,10,20)(6,21,46,59,28,54,15,34)(7,35,43,18,29,60,12,51)(8,52,48,40,30,19,9,57)>;
G:=Group( (1,47,27,12)(2,44,28,9)(3,41,29,14)(4,46,30,11)(5,43,31,16)(6,48,32,13)(7,45,25,10)(8,42,26,15)(17,36,54,57)(18,33,55,62)(19,38,56,59)(20,35,49,64)(21,40,50,61)(22,37,51,58)(23,34,52,63)(24,39,53,60), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(25,31,29,27)(26,32,30,28)(33,60,37,64)(34,61,38,57)(35,62,39,58)(36,63,40,59)(41,47,45,43)(42,48,46,44), (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), (1,58,45,49,31,33,14,24)(2,17,42,63,32,50,11,38)(3,39,47,22,25,64,16,55)(4,56,44,36,26,23,13,61)(5,62,41,53,27,37,10,20)(6,21,46,59,28,54,15,34)(7,35,43,18,29,60,12,51)(8,52,48,40,30,19,9,57) );
G=PermutationGroup([[(1,47,27,12),(2,44,28,9),(3,41,29,14),(4,46,30,11),(5,43,31,16),(6,48,32,13),(7,45,25,10),(8,42,26,15),(17,36,54,57),(18,33,55,62),(19,38,56,59),(20,35,49,64),(21,40,50,61),(22,37,51,58),(23,34,52,63),(24,39,53,60)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,52,21,56),(18,53,22,49),(19,54,23,50),(20,55,24,51),(25,31,29,27),(26,32,30,28),(33,60,37,64),(34,61,38,57),(35,62,39,58),(36,63,40,59),(41,47,45,43),(42,48,46,44)], [(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)], [(1,58,45,49,31,33,14,24),(2,17,42,63,32,50,11,38),(3,39,47,22,25,64,16,55),(4,56,44,36,26,23,13,61),(5,62,41,53,27,37,10,20),(6,21,46,59,28,54,15,34),(7,35,43,18,29,60,12,51),(8,52,48,40,30,19,9,57)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | Q8 | C4≀C2 | C8.C4 | M4(2)⋊4C4 |
| kernel | C42.370D4 | C2×C8⋊C4 | C42.12C4 | C42.6C4 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C42 | C22×C4 | C22×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 8 | 8 | 2 |
Matrix representation of C42.370D4 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 2 | 14 | 0 | 0 |
| 14 | 15 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 3 | 1 |
| 14 | 14 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 2 | 15 |
| 0 | 0 | 4 | 15 |
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[2,14,0,0,14,15,0,0,0,0,4,3,0,0,0,1],[14,3,0,0,14,14,0,0,0,0,2,4,0,0,15,15] >;
C42.370D4 in GAP, Magma, Sage, TeX
C_4^2._{370}D_4 % in TeX
G:=Group("C4^2.370D4"); // GroupNames label
G:=SmallGroup(128,34);
// by ID
G=gap.SmallGroup(128,34);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b*a=a*b,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*b^2*c^3>;
// generators/relations