p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.695C23, C4.1172- 1+4, (C8×Q8)⋊3C2, (C2×Q8)⋊7C8, Q8.8(C2×C8), Q8○2(C22⋊C8), C2.8(C23×C8), (C4×Q8).30C4, C4.20(C22×C8), (C4×C8).30C22, C4⋊C8.376C22, (C2×C4).679C24, (C2×C8).482C23, C42.227(C2×C4), (C22×Q8).32C4, C2.4(Q8○M4(2)), C22.15(C22×C8), C22.44(C23×C4), (C4×Q8).333C22, C22⋊C8.245C22, (C2×C42).786C22, C23.230(C22×C4), C42.12C4.32C2, (C22×C4).1283C23, C2.3(C23.32C23), C22⋊C8○(C4×Q8), (C2×C4⋊C4).79C4, (C2×C4×Q8).45C2, (C2×C4).32(C2×C8), C4⋊C4.251(C2×C4), (C2×Q8).229(C2×C4), (C22×C4).140(C2×C4), (C2×C4).475(C22×C4), SmallGroup(128,1714)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.695C23
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >
Subgroups: 228 in 198 conjugacy classes, 174 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C42.12C4, C8×Q8, C2×C4×Q8, C42.695C23
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, 2- 1+4, C23.32C23, C23×C8, Q8○M4(2), C42.695C23
►On 64 points
(1 35 55 14)(2 36 56 15)(3 37 49 16)(4 38 50 9)(5 39 51 10)(6 40 52 11)(7 33 53 12)(8 34 54 13)(17 41 25 57)(18 42 26 58)(19 43 27 59)(20 44 28 60)(21 45 29 61)(22 46 30 62)(23 47 31 63)(24 48 32 64)
(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 31 55 23)(2 32 56 24)(3 25 49 17)(4 26 50 18)(5 27 51 19)(6 28 52 20)(7 29 53 21)(8 30 54 22)(9 58 38 42)(10 59 39 43)(11 60 40 44)(12 61 33 45)(13 62 34 46)(14 63 35 47)(15 64 36 48)(16 57 37 41)
(1 14 55 35)(2 36 56 15)(3 16 49 37)(4 38 50 9)(5 10 51 39)(6 40 52 11)(7 12 53 33)(8 34 54 13)(17 57 25 41)(18 42 26 58)(19 59 27 43)(20 44 28 60)(21 61 29 45)(22 46 30 62)(23 63 31 47)(24 48 32 64)
G:=sub<Sym(64)| (1,35,55,14)(2,36,56,15)(3,37,49,16)(4,38,50,9)(5,39,51,10)(6,40,52,11)(7,33,53,12)(8,34,54,13)(17,41,25,57)(18,42,26,58)(19,43,27,59)(20,44,28,60)(21,45,29,61)(22,46,30,62)(23,47,31,63)(24,48,32,64), (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), (1,31,55,23)(2,32,56,24)(3,25,49,17)(4,26,50,18)(5,27,51,19)(6,28,52,20)(7,29,53,21)(8,30,54,22)(9,58,38,42)(10,59,39,43)(11,60,40,44)(12,61,33,45)(13,62,34,46)(14,63,35,47)(15,64,36,48)(16,57,37,41), (1,14,55,35)(2,36,56,15)(3,16,49,37)(4,38,50,9)(5,10,51,39)(6,40,52,11)(7,12,53,33)(8,34,54,13)(17,57,25,41)(18,42,26,58)(19,59,27,43)(20,44,28,60)(21,61,29,45)(22,46,30,62)(23,63,31,47)(24,48,32,64)>;
G:=Group( (1,35,55,14)(2,36,56,15)(3,37,49,16)(4,38,50,9)(5,39,51,10)(6,40,52,11)(7,33,53,12)(8,34,54,13)(17,41,25,57)(18,42,26,58)(19,43,27,59)(20,44,28,60)(21,45,29,61)(22,46,30,62)(23,47,31,63)(24,48,32,64), (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), (1,31,55,23)(2,32,56,24)(3,25,49,17)(4,26,50,18)(5,27,51,19)(6,28,52,20)(7,29,53,21)(8,30,54,22)(9,58,38,42)(10,59,39,43)(11,60,40,44)(12,61,33,45)(13,62,34,46)(14,63,35,47)(15,64,36,48)(16,57,37,41), (1,14,55,35)(2,36,56,15)(3,16,49,37)(4,38,50,9)(5,10,51,39)(6,40,52,11)(7,12,53,33)(8,34,54,13)(17,57,25,41)(18,42,26,58)(19,59,27,43)(20,44,28,60)(21,61,29,45)(22,46,30,62)(23,63,31,47)(24,48,32,64) );
G=PermutationGroup([[(1,35,55,14),(2,36,56,15),(3,37,49,16),(4,38,50,9),(5,39,51,10),(6,40,52,11),(7,33,53,12),(8,34,54,13),(17,41,25,57),(18,42,26,58),(19,43,27,59),(20,44,28,60),(21,45,29,61),(22,46,30,62),(23,47,31,63),(24,48,32,64)], [(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)], [(1,31,55,23),(2,32,56,24),(3,25,49,17),(4,26,50,18),(5,27,51,19),(6,28,52,20),(7,29,53,21),(8,30,54,22),(9,58,38,42),(10,59,39,43),(11,60,40,44),(12,61,33,45),(13,62,34,46),(14,63,35,47),(15,64,36,48),(16,57,37,41)], [(1,14,55,35),(2,36,56,15),(3,16,49,37),(4,38,50,9),(5,10,51,39),(6,40,52,11),(7,12,53,33),(8,34,54,13),(17,57,25,41),(18,42,26,58),(19,59,27,43),(20,44,28,60),(21,61,29,45),(22,46,30,62),(23,63,31,47),(24,48,32,64)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4AD | 8A | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | 2- 1+4 | Q8○M4(2) |
| kernel | C42.695C23 | C42.12C4 | C8×Q8 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C4 | C2 |
| # reps | 1 | 6 | 8 | 1 | 6 | 8 | 2 | 32 | 2 | 2 |
Matrix representation of C42.695C23 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 5 | 12 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 15 | 0 |
| 0 | 0 | 2 | 0 | 15 |
| 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 7 | 7 | 5 | 5 |
| 0 | 7 | 10 | 5 | 12 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,5,5,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,5,12],[13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[9,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,15,0,15,0,0,0,15,0,15],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,12,12,7,7,0,12,5,7,10,0,0,0,5,5,0,0,0,5,12] >;
C42.695C23 in GAP, Magma, Sage, TeX
C_4^2._{695}C_2^3 % in TeX
G:=Group("C4^2.695C2^3"); // GroupNames label
G:=SmallGroup(128,1714);
// by ID
G=gap.SmallGroup(128,1714);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,219,100,675,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b,d^2=e^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=a^2*d>;
// generators/relations