p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.220D4, C42.334C23, (C2×Q8)⋊15Q8, Q8.3(C2×Q8), Q8⋊Q8⋊1C2, Q8.Q8⋊12C2, C4.Q16⋊18C2, C4⋊C8.42C22, C4⋊C4.41C23, (C2×C8).25C23, C4.29(C22×Q8), C4.Q8.8C22, (C2×C4).276C24, (C22×C4).800D4, C23.658(C2×D4), C4⋊Q8.261C22, C2.D8.79C22, C4.97(C8.C22), C4⋊M4(2).2C2, C4.107(C22⋊Q8), (C2×Q8).365C23, (C4×Q8).297C22, (C22×C4).995C23, (C2×C42).822C22, Q8⋊C4.24C22, C23.38D4.2C2, C22.536(C22×D4), C22.46(C22⋊Q8), C2.19(D8⋊C22), M4(2)⋊C4.10C2, (C2×M4(2)).65C22, C42.C2.103C22, (C22×Q8).474C22, C42⋊C2.117C22, C23.37C23.27C2, (C2×C4×Q8).52C2, C4.86(C2×C4○D4), (C2×C4).100(C2×Q8), C2.57(C2×C22⋊Q8), (C2×C4).1437(C2×D4), C2.23(C2×C8.C22), (C2×C4).293(C4○D4), (C2×C4⋊C4).923C22, SmallGroup(128,1810)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.220D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2c3 >
Subgroups: 300 in 188 conjugacy classes, 102 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C22×Q8, C23.38D4, C4⋊M4(2), M4(2)⋊C4, Q8⋊Q8, C4.Q16, Q8.Q8, C2×C4×Q8, C23.37C23, C42.220D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C8.C22, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C8.C22, D8⋊C22, C42.220D4
►On 64 points
(1 40 30 47)(2 48 31 33)(3 34 32 41)(4 42 25 35)(5 36 26 43)(6 44 27 37)(7 38 28 45)(8 46 29 39)(9 20 50 58)(10 59 51 21)(11 22 52 60)(12 61 53 23)(13 24 54 62)(14 63 55 17)(15 18 56 64)(16 57 49 19)
(1 32 5 28)(2 29 6 25)(3 26 7 30)(4 31 8 27)(9 56 13 52)(10 53 14 49)(11 50 15 54)(12 55 16 51)(17 57 21 61)(18 62 22 58)(19 59 23 63)(20 64 24 60)(33 46 37 42)(34 43 38 47)(35 48 39 44)(36 45 40 41)
(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 9 30 50)(2 53 31 12)(3 15 32 56)(4 51 25 10)(5 13 26 54)(6 49 27 16)(7 11 28 52)(8 55 29 14)(17 46 63 39)(18 34 64 41)(19 44 57 37)(20 40 58 47)(21 42 59 35)(22 38 60 45)(23 48 61 33)(24 36 62 43)
G:=sub<Sym(64)| (1,40,30,47)(2,48,31,33)(3,34,32,41)(4,42,25,35)(5,36,26,43)(6,44,27,37)(7,38,28,45)(8,46,29,39)(9,20,50,58)(10,59,51,21)(11,22,52,60)(12,61,53,23)(13,24,54,62)(14,63,55,17)(15,18,56,64)(16,57,49,19), (1,32,5,28)(2,29,6,25)(3,26,7,30)(4,31,8,27)(9,56,13,52)(10,53,14,49)(11,50,15,54)(12,55,16,51)(17,57,21,61)(18,62,22,58)(19,59,23,63)(20,64,24,60)(33,46,37,42)(34,43,38,47)(35,48,39,44)(36,45,40,41), (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), (1,9,30,50)(2,53,31,12)(3,15,32,56)(4,51,25,10)(5,13,26,54)(6,49,27,16)(7,11,28,52)(8,55,29,14)(17,46,63,39)(18,34,64,41)(19,44,57,37)(20,40,58,47)(21,42,59,35)(22,38,60,45)(23,48,61,33)(24,36,62,43)>;
G:=Group( (1,40,30,47)(2,48,31,33)(3,34,32,41)(4,42,25,35)(5,36,26,43)(6,44,27,37)(7,38,28,45)(8,46,29,39)(9,20,50,58)(10,59,51,21)(11,22,52,60)(12,61,53,23)(13,24,54,62)(14,63,55,17)(15,18,56,64)(16,57,49,19), (1,32,5,28)(2,29,6,25)(3,26,7,30)(4,31,8,27)(9,56,13,52)(10,53,14,49)(11,50,15,54)(12,55,16,51)(17,57,21,61)(18,62,22,58)(19,59,23,63)(20,64,24,60)(33,46,37,42)(34,43,38,47)(35,48,39,44)(36,45,40,41), (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), (1,9,30,50)(2,53,31,12)(3,15,32,56)(4,51,25,10)(5,13,26,54)(6,49,27,16)(7,11,28,52)(8,55,29,14)(17,46,63,39)(18,34,64,41)(19,44,57,37)(20,40,58,47)(21,42,59,35)(22,38,60,45)(23,48,61,33)(24,36,62,43) );
G=PermutationGroup([[(1,40,30,47),(2,48,31,33),(3,34,32,41),(4,42,25,35),(5,36,26,43),(6,44,27,37),(7,38,28,45),(8,46,29,39),(9,20,50,58),(10,59,51,21),(11,22,52,60),(12,61,53,23),(13,24,54,62),(14,63,55,17),(15,18,56,64),(16,57,49,19)], [(1,32,5,28),(2,29,6,25),(3,26,7,30),(4,31,8,27),(9,56,13,52),(10,53,14,49),(11,50,15,54),(12,55,16,51),(17,57,21,61),(18,62,22,58),(19,59,23,63),(20,64,24,60),(33,46,37,42),(34,43,38,47),(35,48,39,44),(36,45,40,41)], [(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)], [(1,9,30,50),(2,53,31,12),(3,15,32,56),(4,51,25,10),(5,13,26,54),(6,49,27,16),(7,11,28,52),(8,55,29,14),(17,46,63,39),(18,34,64,41),(19,44,57,37),(20,40,58,47),(21,42,59,35),(22,38,60,45),(23,48,61,33),(24,36,62,43)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C8.C22 | D8⋊C22 |
| kernel | C42.220D4 | C23.38D4 | C4⋊M4(2) | M4(2)⋊C4 | Q8⋊Q8 | C4.Q16 | Q8.Q8 | C2×C4×Q8 | C23.37C23 | C42 | C22×C4 | C2×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.220D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 | 4 | 0 |
| 0 | 0 | 8 | 0 | 0 | 13 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 15 |
| 0 | 0 | 14 | 16 | 2 | 0 |
| 0 | 0 | 10 | 14 | 1 | 14 |
| 0 | 0 | 9 | 14 | 11 | 15 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 15 | 0 |
| 0 | 0 | 2 | 11 | 0 | 15 |
| 0 | 0 | 5 | 7 | 14 | 16 |
| 0 | 0 | 14 | 10 | 15 | 6 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,8,0,0,0,13,13,0,0,0,0,0,4,0,0,0,0,0,0,13],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,2,14,10,9,0,0,11,16,14,14,0,0,0,2,1,11,0,0,15,0,14,15],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,3,2,5,14,0,0,1,11,7,10,0,0,15,0,14,15,0,0,0,15,16,6] >;
C42.220D4 in GAP, Magma, Sage, TeX
C_4^2._{220}D_4 % in TeX
G:=Group("C4^2.220D4"); // GroupNames label
G:=SmallGroup(128,1810);
// by ID
G=gap.SmallGroup(128,1810);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,352,2019,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations