p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.235D4, C42.351C23, Q8.Q8⋊18C2, Q16⋊C4⋊8C2, C4⋊C4.70C23, C4⋊C8.53C22, (C2×C8).44C23, Q8.8(C4○D4), Q8⋊D4.1C2, (C2×C4).315C24, C42.6C4⋊8C2, Q8.D4⋊19C2, C22⋊Q16⋊15C2, (C4×D4).79C22, (C2×D4).93C23, C23.677(C2×D4), (C22×C4).455D4, SD16⋊C4⋊11C2, (C4×Q8).76C22, C2.D8.90C22, C8⋊C4.10C22, C4.Q8.19C22, C22⋊C8.28C22, (C2×Q8).379C23, (C2×Q16).59C22, D4⋊C4.34C22, C23.48D4⋊16C2, C4⋊D4.169C22, (C2×C42).842C22, Q8⋊C4.35C22, (C2×SD16).16C22, C23.46D4.1C2, C22.575(C22×D4), C22⋊Q8.174C22, C2.34(D8⋊C22), (C22×C4).1031C23, C4.4D4.131C22, C22.19(C8.C22), C42.C2.108C22, (C22×Q8).480C22, C2.116(C22.19C24), C23.36C23.21C2, (C2×C4×Q8)⋊40C2, C4.200(C2×C4○D4), (C2×C4).1223(C2×D4), C2.36(C2×C8.C22), (C2×C4⋊C4).941C22, SmallGroup(128,1849)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.235D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b, dcd=c3 >
Subgroups: 332 in 193 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×SD16, C2×Q16, C22×Q8, C42.6C4, SD16⋊C4, Q16⋊C4, Q8⋊D4, C22⋊Q16, Q8.D4, Q8.Q8, C23.46D4, C23.48D4, C2×C4×Q8, C23.36C23, C42.235D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, C22.19C24, C2×C8.C22, D8⋊C22, C42.235D4
►On 64 points
(1 63 5 59)(2 49 6 53)(3 57 7 61)(4 51 8 55)(9 19 13 23)(10 45 14 41)(11 21 15 17)(12 47 16 43)(18 36 22 40)(20 38 24 34)(25 52 29 56)(26 60 30 64)(27 54 31 50)(28 62 32 58)(33 44 37 48)(35 46 39 42)
(1 37 29 13)(2 34 30 10)(3 39 31 15)(4 36 32 12)(5 33 25 9)(6 38 26 14)(7 35 27 11)(8 40 28 16)(17 57 42 50)(18 62 43 55)(19 59 44 52)(20 64 45 49)(21 61 46 54)(22 58 47 51)(23 63 48 56)(24 60 41 53)
(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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 46)(18 41)(19 44)(20 47)(21 42)(22 45)(23 48)(24 43)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(49 62)(50 57)(51 60)(52 63)(53 58)(54 61)(55 64)(56 59)
G:=sub<Sym(64)| (1,63,5,59)(2,49,6,53)(3,57,7,61)(4,51,8,55)(9,19,13,23)(10,45,14,41)(11,21,15,17)(12,47,16,43)(18,36,22,40)(20,38,24,34)(25,52,29,56)(26,60,30,64)(27,54,31,50)(28,62,32,58)(33,44,37,48)(35,46,39,42), (1,37,29,13)(2,34,30,10)(3,39,31,15)(4,36,32,12)(5,33,25,9)(6,38,26,14)(7,35,27,11)(8,40,28,16)(17,57,42,50)(18,62,43,55)(19,59,44,52)(20,64,45,49)(21,61,46,54)(22,58,47,51)(23,63,48,56)(24,60,41,53), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(49,62)(50,57)(51,60)(52,63)(53,58)(54,61)(55,64)(56,59)>;
G:=Group( (1,63,5,59)(2,49,6,53)(3,57,7,61)(4,51,8,55)(9,19,13,23)(10,45,14,41)(11,21,15,17)(12,47,16,43)(18,36,22,40)(20,38,24,34)(25,52,29,56)(26,60,30,64)(27,54,31,50)(28,62,32,58)(33,44,37,48)(35,46,39,42), (1,37,29,13)(2,34,30,10)(3,39,31,15)(4,36,32,12)(5,33,25,9)(6,38,26,14)(7,35,27,11)(8,40,28,16)(17,57,42,50)(18,62,43,55)(19,59,44,52)(20,64,45,49)(21,61,46,54)(22,58,47,51)(23,63,48,56)(24,60,41,53), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(49,62)(50,57)(51,60)(52,63)(53,58)(54,61)(55,64)(56,59) );
G=PermutationGroup([[(1,63,5,59),(2,49,6,53),(3,57,7,61),(4,51,8,55),(9,19,13,23),(10,45,14,41),(11,21,15,17),(12,47,16,43),(18,36,22,40),(20,38,24,34),(25,52,29,56),(26,60,30,64),(27,54,31,50),(28,62,32,58),(33,44,37,48),(35,46,39,42)], [(1,37,29,13),(2,34,30,10),(3,39,31,15),(4,36,32,12),(5,33,25,9),(6,38,26,14),(7,35,27,11),(8,40,28,16),(17,57,42,50),(18,62,43,55),(19,59,44,52),(20,64,45,49),(21,61,46,54),(22,58,47,51),(23,63,48,56),(24,60,41,53)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,46),(18,41),(19,44),(20,47),(21,42),(22,45),(23,48),(24,43),(26,28),(27,31),(30,32),(33,37),(34,40),(36,38),(49,62),(50,57),(51,60),(52,63),(53,58),(54,61),(55,64),(56,59)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8.C22 | D8⋊C22 |
| kernel | C42.235D4 | C42.6C4 | SD16⋊C4 | Q16⋊C4 | Q8⋊D4 | C22⋊Q16 | Q8.D4 | Q8.Q8 | C23.46D4 | C23.48D4 | C2×C4×Q8 | C23.36C23 | C42 | C22×C4 | Q8 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.235D4 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 16 | 16 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 15 |
| 0 | 0 | 16 | 0 | 16 | 1 |
| 0 | 0 | 16 | 16 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 9 | 11 | 6 |
| 0 | 0 | 7 | 0 | 6 | 0 |
| 0 | 0 | 8 | 1 | 7 | 10 |
| 0 | 0 | 10 | 16 | 1 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,2,1,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,16,16,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,15,1,1,16],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,3,7,8,10,0,0,9,0,1,16,0,0,11,6,7,1,0,0,6,0,10,7],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C42.235D4 in GAP, Magma, Sage, TeX
C_4^2._{235}D_4 % in TeX
G:=Group("C4^2.235D4"); // GroupNames label
G:=SmallGroup(128,1849);
// by ID
G=gap.SmallGroup(128,1849);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,521,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations