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, C23.658(C2×D4), (C22×C4).800D4, 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
Subgroups: 300 in 188 conjugacy classes, 102 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×13], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×17], Q8 [×4], Q8 [×10], C23, C42 [×4], C42 [×6], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×5], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C23.38D4 [×2], C4⋊M4(2), M4(2)⋊C4 [×2], Q8⋊Q8 [×2], C4.Q16 [×2], Q8.Q8 [×4], C2×C4×Q8, C23.37C23, C42.220D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C8.C22 [×2], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C8.C22, D8⋊C22, C42.220D4
Generators and relations
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 >
►On 64 points
(1 38 26 47)(2 48 27 39)(3 40 28 41)(4 42 29 33)(5 34 30 43)(6 44 31 35)(7 36 32 45)(8 46 25 37)(9 22 54 62)(10 63 55 23)(11 24 56 64)(12 57 49 17)(13 18 50 58)(14 59 51 19)(15 20 52 60)(16 61 53 21)
(1 28 5 32)(2 25 6 29)(3 30 7 26)(4 27 8 31)(9 52 13 56)(10 49 14 53)(11 54 15 50)(12 51 16 55)(17 59 21 63)(18 64 22 60)(19 61 23 57)(20 58 24 62)(33 48 37 44)(34 45 38 41)(35 42 39 46)(36 47 40 43)
(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 13 26 50)(2 53 27 16)(3 11 28 56)(4 51 29 14)(5 9 30 54)(6 49 31 12)(7 15 32 52)(8 55 25 10)(17 44 57 35)(18 38 58 47)(19 42 59 33)(20 36 60 45)(21 48 61 39)(22 34 62 43)(23 46 63 37)(24 40 64 41)
G:=sub<Sym(64)| (1,38,26,47)(2,48,27,39)(3,40,28,41)(4,42,29,33)(5,34,30,43)(6,44,31,35)(7,36,32,45)(8,46,25,37)(9,22,54,62)(10,63,55,23)(11,24,56,64)(12,57,49,17)(13,18,50,58)(14,59,51,19)(15,20,52,60)(16,61,53,21), (1,28,5,32)(2,25,6,29)(3,30,7,26)(4,27,8,31)(9,52,13,56)(10,49,14,53)(11,54,15,50)(12,51,16,55)(17,59,21,63)(18,64,22,60)(19,61,23,57)(20,58,24,62)(33,48,37,44)(34,45,38,41)(35,42,39,46)(36,47,40,43), (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), (1,13,26,50)(2,53,27,16)(3,11,28,56)(4,51,29,14)(5,9,30,54)(6,49,31,12)(7,15,32,52)(8,55,25,10)(17,44,57,35)(18,38,58,47)(19,42,59,33)(20,36,60,45)(21,48,61,39)(22,34,62,43)(23,46,63,37)(24,40,64,41)>;
G:=Group( (1,38,26,47)(2,48,27,39)(3,40,28,41)(4,42,29,33)(5,34,30,43)(6,44,31,35)(7,36,32,45)(8,46,25,37)(9,22,54,62)(10,63,55,23)(11,24,56,64)(12,57,49,17)(13,18,50,58)(14,59,51,19)(15,20,52,60)(16,61,53,21), (1,28,5,32)(2,25,6,29)(3,30,7,26)(4,27,8,31)(9,52,13,56)(10,49,14,53)(11,54,15,50)(12,51,16,55)(17,59,21,63)(18,64,22,60)(19,61,23,57)(20,58,24,62)(33,48,37,44)(34,45,38,41)(35,42,39,46)(36,47,40,43), (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), (1,13,26,50)(2,53,27,16)(3,11,28,56)(4,51,29,14)(5,9,30,54)(6,49,31,12)(7,15,32,52)(8,55,25,10)(17,44,57,35)(18,38,58,47)(19,42,59,33)(20,36,60,45)(21,48,61,39)(22,34,62,43)(23,46,63,37)(24,40,64,41) );
G=PermutationGroup([(1,38,26,47),(2,48,27,39),(3,40,28,41),(4,42,29,33),(5,34,30,43),(6,44,31,35),(7,36,32,45),(8,46,25,37),(9,22,54,62),(10,63,55,23),(11,24,56,64),(12,57,49,17),(13,18,50,58),(14,59,51,19),(15,20,52,60),(16,61,53,21)], [(1,28,5,32),(2,25,6,29),(3,30,7,26),(4,27,8,31),(9,52,13,56),(10,49,14,53),(11,54,15,50),(12,51,16,55),(17,59,21,63),(18,64,22,60),(19,61,23,57),(20,58,24,62),(33,48,37,44),(34,45,38,41),(35,42,39,46),(36,47,40,43)], [(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)], [(1,13,26,50),(2,53,27,16),(3,11,28,56),(4,51,29,14),(5,9,30,54),(6,49,31,12),(7,15,32,52),(8,55,25,10),(17,44,57,35),(18,38,58,47),(19,42,59,33),(20,36,60,45),(21,48,61,39),(22,34,62,43),(23,46,63,37),(24,40,64,41)])
Matrix representation ►G ⊆ 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] >;
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 |
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