p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.114D4, M4(2).28D4, C22.59(C4×D4), C4.18(C4⋊1D4), C4⋊1(C4.10D4), C4.79(C4⋊D4), C4.64(C4.4D4), (C4×M4(2)).25C2, C4⋊M4(2).32C2, (C2×C42).320C22, (C22×C4).700C23, C23.198(C22×C4), (C22×Q8).36C22, (C2×M4(2)).209C22, C2.18(C24.3C22), (C2×C4⋊C4).24C4, (C2×C4⋊Q8).11C2, (C2×C4).65(C4○D4), (C2×C4).1352(C2×D4), (C22×C4).24(C2×C4), (C2×C4.10D4).9C2, C2.26(C2×C4.10D4), (C2×C4).258(C22⋊C4), C22.288(C2×C22⋊C4), SmallGroup(128,698)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.114D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b-1c3 >
Subgroups: 260 in 146 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C4×C8, C8⋊C4, C4.10D4, C4⋊C8, C2×C42, C2×C4⋊C4, C4⋊Q8, C2×M4(2), C22×Q8, C4×M4(2), C2×C4.10D4, C4⋊M4(2), C2×C4⋊Q8, C42.114D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4.10D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22, C2×C4.10D4, C42.114D4
►On 64 points
(1 31 13 39)(2 32 14 40)(3 25 15 33)(4 26 16 34)(5 27 9 35)(6 28 10 36)(7 29 11 37)(8 30 12 38)(17 49 45 58)(18 50 46 59)(19 51 47 60)(20 52 48 61)(21 53 41 62)(22 54 42 63)(23 55 43 64)(24 56 44 57)
(1 11 5 15)(2 16 6 12)(3 13 7 9)(4 10 8 14)(17 43 21 47)(18 48 22 44)(19 45 23 41)(20 42 24 46)(25 39 29 35)(26 36 30 40)(27 33 31 37)(28 38 32 34)(49 64 53 60)(50 61 54 57)(51 58 55 62)(52 63 56 59)
(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 49 5 53)(2 63 6 59)(3 55 7 51)(4 61 8 57)(9 62 13 58)(10 50 14 54)(11 60 15 64)(12 56 16 52)(17 27 21 31)(18 40 22 36)(19 25 23 29)(20 38 24 34)(26 48 30 44)(28 46 32 42)(33 43 37 47)(35 41 39 45)
G:=sub<Sym(64)| (1,31,13,39)(2,32,14,40)(3,25,15,33)(4,26,16,34)(5,27,9,35)(6,28,10,36)(7,29,11,37)(8,30,12,38)(17,49,45,58)(18,50,46,59)(19,51,47,60)(20,52,48,61)(21,53,41,62)(22,54,42,63)(23,55,43,64)(24,56,44,57), (1,11,5,15)(2,16,6,12)(3,13,7,9)(4,10,8,14)(17,43,21,47)(18,48,22,44)(19,45,23,41)(20,42,24,46)(25,39,29,35)(26,36,30,40)(27,33,31,37)(28,38,32,34)(49,64,53,60)(50,61,54,57)(51,58,55,62)(52,63,56,59), (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), (1,49,5,53)(2,63,6,59)(3,55,7,51)(4,61,8,57)(9,62,13,58)(10,50,14,54)(11,60,15,64)(12,56,16,52)(17,27,21,31)(18,40,22,36)(19,25,23,29)(20,38,24,34)(26,48,30,44)(28,46,32,42)(33,43,37,47)(35,41,39,45)>;
G:=Group( (1,31,13,39)(2,32,14,40)(3,25,15,33)(4,26,16,34)(5,27,9,35)(6,28,10,36)(7,29,11,37)(8,30,12,38)(17,49,45,58)(18,50,46,59)(19,51,47,60)(20,52,48,61)(21,53,41,62)(22,54,42,63)(23,55,43,64)(24,56,44,57), (1,11,5,15)(2,16,6,12)(3,13,7,9)(4,10,8,14)(17,43,21,47)(18,48,22,44)(19,45,23,41)(20,42,24,46)(25,39,29,35)(26,36,30,40)(27,33,31,37)(28,38,32,34)(49,64,53,60)(50,61,54,57)(51,58,55,62)(52,63,56,59), (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), (1,49,5,53)(2,63,6,59)(3,55,7,51)(4,61,8,57)(9,62,13,58)(10,50,14,54)(11,60,15,64)(12,56,16,52)(17,27,21,31)(18,40,22,36)(19,25,23,29)(20,38,24,34)(26,48,30,44)(28,46,32,42)(33,43,37,47)(35,41,39,45) );
G=PermutationGroup([[(1,31,13,39),(2,32,14,40),(3,25,15,33),(4,26,16,34),(5,27,9,35),(6,28,10,36),(7,29,11,37),(8,30,12,38),(17,49,45,58),(18,50,46,59),(19,51,47,60),(20,52,48,61),(21,53,41,62),(22,54,42,63),(23,55,43,64),(24,56,44,57)], [(1,11,5,15),(2,16,6,12),(3,13,7,9),(4,10,8,14),(17,43,21,47),(18,48,22,44),(19,45,23,41),(20,42,24,46),(25,39,29,35),(26,36,30,40),(27,33,31,37),(28,38,32,34),(49,64,53,60),(50,61,54,57),(51,58,55,62),(52,63,56,59)], [(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)], [(1,49,5,53),(2,63,6,59),(3,55,7,51),(4,61,8,57),(9,62,13,58),(10,50,14,54),(11,60,15,64),(12,56,16,52),(17,27,21,31),(18,40,22,36),(19,25,23,29),(20,38,24,34),(26,48,30,44),(28,46,32,42),(33,43,37,47),(35,41,39,45)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| 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 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C4.10D4 |
| kernel | C42.114D4 | C4×M4(2) | C2×C4.10D4 | C4⋊M4(2) | C2×C4⋊Q8 | C2×C4⋊C4 | C42 | M4(2) | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 4 | 4 | 4 | 4 |
Matrix representation of C42.114D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 10 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 7 |
| 0 | 0 | 0 | 0 | 7 | 1 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,10,0,0,0,0,10,16,0,0,0,0,0,0,16,7,0,0,0,0,7,1] >;
C42.114D4 in GAP, Magma, Sage, TeX
C_4^2._{114}D_4 % in TeX
G:=Group("C4^2.114D4"); // GroupNames label
G:=SmallGroup(128,698);
// by ID
G=gap.SmallGroup(128,698);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,100,2019,1018,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^3>;
// generators/relations