p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.397D4, C42.598C23, D4⋊C8⋊1C2, C4○D4⋊2C8, D4⋊5(C2×C8), Q8⋊C8⋊1C2, Q8⋊5(C2×C8), (C4×D4).13C4, (C4×C8).1C22, C4.4(C22×C8), (C4×Q8).13C4, C42.51(C2×C4), C4.4(C2×M4(2)), C4⋊C8.245C22, C4.23(C22⋊C8), (C22×C4).654D4, (C2×C4).15M4(2), C4.129(C8⋊C22), C42⋊C2.16C4, C42.12C4⋊6C2, (C4×D4).262C22, C22.3(C22⋊C8), (C4×Q8).249C22, C4.123(C8.C22), C23.93(C22⋊C4), (C2×C42).154C22, C2.1(C42⋊C22), C2.1(C23.36D4), (C2×C4⋊C8)⋊2C2, (C2×C4).16(C2×C8), (C4×C4○D4).2C2, C4⋊C4.176(C2×C4), (C2×C4○D4).14C4, C2.13(C2×C22⋊C8), (C2×D4).190(C2×C4), (C2×C4).1441(C2×D4), (C2×Q8).173(C2×C4), (C22×C4).176(C2×C4), (C2×C4).303(C22×C4), C22.97(C2×C22⋊C4), (C2×C4).351(C22⋊C4), SmallGroup(128,209)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.397D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 236 in 132 conjugacy classes, 62 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×C4○D4, D4⋊C8, Q8⋊C8, C2×C4⋊C8, C42.12C4, C4×C4○D4, C42.397D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C8⋊C22, C8.C22, C2×C22⋊C8, C23.36D4, C42⋊C22, C42.397D4
►On 64 points
(1 13 55 27)(2 28 56 14)(3 15 49 29)(4 30 50 16)(5 9 51 31)(6 32 52 10)(7 11 53 25)(8 26 54 12)(17 45 61 37)(18 38 62 46)(19 47 63 39)(20 40 64 48)(21 41 57 33)(22 34 58 42)(23 43 59 35)(24 36 60 44)
(1 21 51 61)(2 22 52 62)(3 23 53 63)(4 24 54 64)(5 17 55 57)(6 18 56 58)(7 19 49 59)(8 20 50 60)(9 45 27 33)(10 46 28 34)(11 47 29 35)(12 48 30 36)(13 41 31 37)(14 42 32 38)(15 43 25 39)(16 44 26 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 20 21 50 51 60 61 8)(2 49 22 59 52 7 62 19)(3 58 23 6 53 18 63 56)(4 5 24 17 54 55 64 57)(9 44 45 26 27 40 33 16)(10 25 46 39 28 15 34 43)(11 38 47 14 29 42 35 32)(12 13 48 41 30 31 36 37)
G:=sub<Sym(64)| (1,13,55,27)(2,28,56,14)(3,15,49,29)(4,30,50,16)(5,9,51,31)(6,32,52,10)(7,11,53,25)(8,26,54,12)(17,45,61,37)(18,38,62,46)(19,47,63,39)(20,40,64,48)(21,41,57,33)(22,34,58,42)(23,43,59,35)(24,36,60,44), (1,21,51,61)(2,22,52,62)(3,23,53,63)(4,24,54,64)(5,17,55,57)(6,18,56,58)(7,19,49,59)(8,20,50,60)(9,45,27,33)(10,46,28,34)(11,47,29,35)(12,48,30,36)(13,41,31,37)(14,42,32,38)(15,43,25,39)(16,44,26,40), (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), (1,20,21,50,51,60,61,8)(2,49,22,59,52,7,62,19)(3,58,23,6,53,18,63,56)(4,5,24,17,54,55,64,57)(9,44,45,26,27,40,33,16)(10,25,46,39,28,15,34,43)(11,38,47,14,29,42,35,32)(12,13,48,41,30,31,36,37)>;
G:=Group( (1,13,55,27)(2,28,56,14)(3,15,49,29)(4,30,50,16)(5,9,51,31)(6,32,52,10)(7,11,53,25)(8,26,54,12)(17,45,61,37)(18,38,62,46)(19,47,63,39)(20,40,64,48)(21,41,57,33)(22,34,58,42)(23,43,59,35)(24,36,60,44), (1,21,51,61)(2,22,52,62)(3,23,53,63)(4,24,54,64)(5,17,55,57)(6,18,56,58)(7,19,49,59)(8,20,50,60)(9,45,27,33)(10,46,28,34)(11,47,29,35)(12,48,30,36)(13,41,31,37)(14,42,32,38)(15,43,25,39)(16,44,26,40), (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), (1,20,21,50,51,60,61,8)(2,49,22,59,52,7,62,19)(3,58,23,6,53,18,63,56)(4,5,24,17,54,55,64,57)(9,44,45,26,27,40,33,16)(10,25,46,39,28,15,34,43)(11,38,47,14,29,42,35,32)(12,13,48,41,30,31,36,37) );
G=PermutationGroup([[(1,13,55,27),(2,28,56,14),(3,15,49,29),(4,30,50,16),(5,9,51,31),(6,32,52,10),(7,11,53,25),(8,26,54,12),(17,45,61,37),(18,38,62,46),(19,47,63,39),(20,40,64,48),(21,41,57,33),(22,34,58,42),(23,43,59,35),(24,36,60,44)], [(1,21,51,61),(2,22,52,62),(3,23,53,63),(4,24,54,64),(5,17,55,57),(6,18,56,58),(7,19,49,59),(8,20,50,60),(9,45,27,33),(10,46,28,34),(11,47,29,35),(12,48,30,36),(13,41,31,37),(14,42,32,38),(15,43,25,39),(16,44,26,40)], [(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)], [(1,20,21,50,51,60,61,8),(2,49,22,59,52,7,62,19),(3,58,23,6,53,18,63,56),(4,5,24,17,54,55,64,57),(9,44,45,26,27,40,33,16),(10,25,46,39,28,15,34,43),(11,38,47,14,29,42,35,32),(12,13,48,41,30,31,36,37)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | D4 | M4(2) | C8⋊C22 | C8.C22 | C42⋊C22 |
| kernel | C42.397D4 | D4⋊C8 | Q8⋊C8 | C2×C4⋊C8 | C42.12C4 | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C42 | C22×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C42.397D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 16 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 4 | 4 | 13 | 13 |
| 0 | 0 | 0 | 13 | 4 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 13 | 13 | 4 | 4 |
| 0 | 0 | 4 | 0 | 13 | 13 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,16,16,0,16,0,0,15,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,13,0,4,13,0,0,0,4,13,4,0,0,4,4,13,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,4,0,0,13,0,13,0,0,0,0,13,4,13,0,0,4,13,4,13] >;
C42.397D4 in GAP, Magma, Sage, TeX
C_4^2._{397}D_4 % in TeX
G:=Group("C4^2.397D4"); // GroupNames label
G:=SmallGroup(128,209);
// by ID
G=gap.SmallGroup(128,209);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,1123,570,136,172]);
// 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=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations