p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.322D4, (C4×C8).21C4, C4.7(C8.C4), (C22×C4).91Q8, C23.81(C2×Q8), C42.321(C2×C4), C4.82(C4.4D4), C4.C42.9C2, C4.60(C42⋊C2), C2.10(C42⋊8C4), C4⋊M4(2).27C2, (C22×C8).480C22, C22.1(C42.C2), (C22×C4).1341C23, (C2×C42).1057C22, (C2×M4(2)).167C22, (C2×C4×C8).19C2, (C2×C4).87(C4⋊C4), (C2×C8).213(C2×C4), C22.99(C2×C4⋊C4), C2.12(C2×C8.C4), (C2×C4).1521(C2×D4), (C2×C4).559(C4○D4), (C2×C4).539(C22×C4), SmallGroup(128,569)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.322D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c3 >
Subgroups: 140 in 94 conjugacy classes, 56 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C4.C42, C2×C4×C8, C4⋊M4(2), C42.322D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C42⋊8C4, C2×C8.C4, C42.322D4
►On 64 points
(1 34 27 15)(2 35 28 16)(3 36 29 9)(4 37 30 10)(5 38 31 11)(6 39 32 12)(7 40 25 13)(8 33 26 14)(17 60 44 55)(18 61 45 56)(19 62 46 49)(20 63 47 50)(21 64 48 51)(22 57 41 52)(23 58 42 53)(24 59 43 54)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 46 21 42)(18 47 22 43)(19 48 23 44)(20 41 24 45)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)
(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 7 60 5 62 3 64)(2 52 8 54 6 56 4 50)(9 48 15 42 13 44 11 46)(10 20 16 22 14 24 12 18)(17 38 19 36 21 34 23 40)(25 55 31 49 29 51 27 53)(26 59 32 61 30 63 28 57)(33 43 39 45 37 47 35 41)
G:=sub<Sym(64)| (1,34,27,15)(2,35,28,16)(3,36,29,9)(4,37,30,10)(5,38,31,11)(6,39,32,12)(7,40,25,13)(8,33,26,14)(17,60,44,55)(18,61,45,56)(19,62,46,49)(20,63,47,50)(21,64,48,51)(22,57,41,52)(23,58,42,53)(24,59,43,54), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,46,21,42)(18,47,22,43)(19,48,23,44)(20,41,24,45)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63), (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), (1,58,7,60,5,62,3,64)(2,52,8,54,6,56,4,50)(9,48,15,42,13,44,11,46)(10,20,16,22,14,24,12,18)(17,38,19,36,21,34,23,40)(25,55,31,49,29,51,27,53)(26,59,32,61,30,63,28,57)(33,43,39,45,37,47,35,41)>;
G:=Group( (1,34,27,15)(2,35,28,16)(3,36,29,9)(4,37,30,10)(5,38,31,11)(6,39,32,12)(7,40,25,13)(8,33,26,14)(17,60,44,55)(18,61,45,56)(19,62,46,49)(20,63,47,50)(21,64,48,51)(22,57,41,52)(23,58,42,53)(24,59,43,54), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,46,21,42)(18,47,22,43)(19,48,23,44)(20,41,24,45)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63), (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), (1,58,7,60,5,62,3,64)(2,52,8,54,6,56,4,50)(9,48,15,42,13,44,11,46)(10,20,16,22,14,24,12,18)(17,38,19,36,21,34,23,40)(25,55,31,49,29,51,27,53)(26,59,32,61,30,63,28,57)(33,43,39,45,37,47,35,41) );
G=PermutationGroup([[(1,34,27,15),(2,35,28,16),(3,36,29,9),(4,37,30,10),(5,38,31,11),(6,39,32,12),(7,40,25,13),(8,33,26,14),(17,60,44,55),(18,61,45,56),(19,62,46,49),(20,63,47,50),(21,64,48,51),(22,57,41,52),(23,58,42,53),(24,59,43,54)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,34,13,38),(10,35,14,39),(11,36,15,40),(12,37,16,33),(17,46,21,42),(18,47,22,43),(19,48,23,44),(20,41,24,45),(49,64,53,60),(50,57,54,61),(51,58,55,62),(52,59,56,63)], [(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)], [(1,58,7,60,5,62,3,64),(2,52,8,54,6,56,4,50),(9,48,15,42,13,44,11,46),(10,20,16,22,14,24,12,18),(17,38,19,36,21,34,23,40),(25,55,31,49,29,51,27,53),(26,59,32,61,30,63,28,57),(33,43,39,45,37,47,35,41)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | C4○D4 | C8.C4 |
| kernel | C42.322D4 | C4.C42 | C2×C4×C8 | C4⋊M4(2) | C4×C8 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 8 | 2 | 2 | 8 | 16 |
Matrix representation of C42.322D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 8 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 8 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 13 | 1 |
| 0 | 0 | 2 | 4 |
| 0 | 15 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 10 |
| 0 | 0 | 15 | 2 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,8,0,0,4,16],[13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[8,0,0,0,0,2,0,0,0,0,13,2,0,0,1,4],[0,15,0,0,15,0,0,0,0,0,15,15,0,0,10,2] >;
C42.322D4 in GAP, Magma, Sage, TeX
C_4^2._{322}D_4 % in TeX
G:=Group("C4^2.322D4"); // GroupNames label
G:=SmallGroup(128,569);
// by ID
G=gap.SmallGroup(128,569);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,248,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations