p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.293C23, (C8×D4)⋊42C2, C8⋊9D4⋊36C2, C8⋊4Q8⋊36C2, C8.86(C4○D4), C4⋊D4.22C4, C22⋊Q8.22C4, C4⋊C8.363C22, (C4×C8).334C22, C42⋊2C2.2C4, (C2×C4).665C24, (C2×C8).429C23, C42.216(C2×C4), C4.4D4.18C4, (C4×D4).62C22, C22.6(C8○D4), C42.C2.18C4, (C4×Q8).60C22, C8○2M4(2)⋊35C2, C22.D4.6C4, C8⋊C4.174C22, C42.12C4⋊50C2, C22⋊C8.234C22, C2.23(Q8○M4(2)), C22.190(C23×C4), (C2×C42).775C22, (C22×C8).447C22, C23.147(C22×C4), C42.7C22⋊24C2, (C22×C4).1279C23, C42⋊C2.308C22, (C2×M4(2)).367C22, C23.36C23.14C2, (C2×C8⋊C4)⋊35C2, C2.47(C4×C4○D4), C2.25(C2×C8○D4), C4⋊C4.165(C2×C4), C8⋊C4○(C22⋊C8), C4.316(C2×C4○D4), (C2×D4).181(C2×C4), C22⋊C4.42(C2×C4), (C2×C4).78(C22×C4), (C2×Q8).121(C2×C4), (C22×C4).349(C2×C4), SmallGroup(128,1700)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.293C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, de=ed >
Subgroups: 252 in 186 conjugacy classes, 130 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×M4(2), C2×C8⋊C4, C8○2M4(2), C42.12C4, C42.7C22, C8×D4, C8⋊9D4, C8⋊4Q8, C23.36C23, C42.293C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C8○D4, C23×C4, C2×C4○D4, C4×C4○D4, C2×C8○D4, Q8○M4(2), C42.293C23
►On 64 points
(1 42 27 18)(2 19 28 43)(3 44 29 20)(4 21 30 45)(5 46 31 22)(6 23 32 47)(7 48 25 24)(8 17 26 41)(9 49 37 62)(10 63 38 50)(11 51 39 64)(12 57 40 52)(13 53 33 58)(14 59 34 54)(15 55 35 60)(16 61 36 56)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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 6 3 8 5 2 7 4)(9 10 11 12 13 14 15 16)(17 42 19 44 21 46 23 48)(18 43 20 45 22 47 24 41)(25 30 27 32 29 26 31 28)(33 34 35 36 37 38 39 40)(49 59 51 61 53 63 55 57)(50 60 52 62 54 64 56 58)
(1 59)(2 55)(3 61)(4 49)(5 63)(6 51)(7 57)(8 53)(9 45)(10 22)(11 47)(12 24)(13 41)(14 18)(15 43)(16 20)(17 33)(19 35)(21 37)(23 39)(25 52)(26 58)(27 54)(28 60)(29 56)(30 62)(31 50)(32 64)(34 42)(36 44)(38 46)(40 48)
G:=sub<Sym(64)| (1,42,27,18)(2,19,28,43)(3,44,29,20)(4,21,30,45)(5,46,31,22)(6,23,32,47)(7,48,25,24)(8,17,26,41)(9,49,37,62)(10,63,38,50)(11,51,39,64)(12,57,40,52)(13,53,33,58)(14,59,34,54)(15,55,35,60)(16,61,36,56), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,6,3,8,5,2,7,4)(9,10,11,12,13,14,15,16)(17,42,19,44,21,46,23,48)(18,43,20,45,22,47,24,41)(25,30,27,32,29,26,31,28)(33,34,35,36,37,38,39,40)(49,59,51,61,53,63,55,57)(50,60,52,62,54,64,56,58), (1,59)(2,55)(3,61)(4,49)(5,63)(6,51)(7,57)(8,53)(9,45)(10,22)(11,47)(12,24)(13,41)(14,18)(15,43)(16,20)(17,33)(19,35)(21,37)(23,39)(25,52)(26,58)(27,54)(28,60)(29,56)(30,62)(31,50)(32,64)(34,42)(36,44)(38,46)(40,48)>;
G:=Group( (1,42,27,18)(2,19,28,43)(3,44,29,20)(4,21,30,45)(5,46,31,22)(6,23,32,47)(7,48,25,24)(8,17,26,41)(9,49,37,62)(10,63,38,50)(11,51,39,64)(12,57,40,52)(13,53,33,58)(14,59,34,54)(15,55,35,60)(16,61,36,56), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,6,3,8,5,2,7,4)(9,10,11,12,13,14,15,16)(17,42,19,44,21,46,23,48)(18,43,20,45,22,47,24,41)(25,30,27,32,29,26,31,28)(33,34,35,36,37,38,39,40)(49,59,51,61,53,63,55,57)(50,60,52,62,54,64,56,58), (1,59)(2,55)(3,61)(4,49)(5,63)(6,51)(7,57)(8,53)(9,45)(10,22)(11,47)(12,24)(13,41)(14,18)(15,43)(16,20)(17,33)(19,35)(21,37)(23,39)(25,52)(26,58)(27,54)(28,60)(29,56)(30,62)(31,50)(32,64)(34,42)(36,44)(38,46)(40,48) );
G=PermutationGroup([[(1,42,27,18),(2,19,28,43),(3,44,29,20),(4,21,30,45),(5,46,31,22),(6,23,32,47),(7,48,25,24),(8,17,26,41),(9,49,37,62),(10,63,38,50),(11,51,39,64),(12,57,40,52),(13,53,33,58),(14,59,34,54),(15,55,35,60),(16,61,36,56)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,6,3,8,5,2,7,4),(9,10,11,12,13,14,15,16),(17,42,19,44,21,46,23,48),(18,43,20,45,22,47,24,41),(25,30,27,32,29,26,31,28),(33,34,35,36,37,38,39,40),(49,59,51,61,53,63,55,57),(50,60,52,62,54,64,56,58)], [(1,59),(2,55),(3,61),(4,49),(5,63),(6,51),(7,57),(8,53),(9,45),(10,22),(11,47),(12,24),(13,41),(14,18),(15,43),(16,20),(17,33),(19,35),(21,37),(23,39),(25,52),(26,58),(27,54),(28,60),(29,56),(30,62),(31,50),(32,64),(34,42),(36,44),(38,46),(40,48)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | C42.293C23 | C2×C8⋊C4 | C8○2M4(2) | C42.12C4 | C42.7C22 | C8×D4 | C8⋊9D4 | C8⋊4Q8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C8 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 2 |
Matrix representation of C42.293C23 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,0,1,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,13,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,15,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
C42.293C23 in GAP, Magma, Sage, TeX
C_4^2._{293}C_2^3 % in TeX
G:=Group("C4^2.293C2^3"); // GroupNames label
G:=SmallGroup(128,1700);
// by ID
G=gap.SmallGroup(128,1700);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^2*c,d*e=e*d>;
// generators/relations