p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.304C23, C4.1212- 1+4, (C8×Q8)⋊35C2, C4⋊Q8.36C4, C8⋊4Q8⋊42C2, C4.41(C8○D4), (C4×C8).31C22, C22⋊Q8.28C4, C4⋊C8.369C22, (C2×C4).683C24, (C2×C8).443C23, C42.231(C2×C4), C42.C2.21C4, (C4×Q8).284C22, C8⋊C4.102C22, C22⋊C8.239C22, C23.108(C22×C4), C22.206(C23×C4), (C22×C4).947C23, (C2×C42).790C22, C42.12C4.48C2, C42⋊C2.89C22, C42.7C22.4C2, C23.37C23.25C2, C2.26(C23.32C23), C2.34(C2×C8○D4), C4⋊C4.170(C2×C4), C22⋊C4.24(C2×C4), (C2×C4).85(C22×C4), (C2×Q8).167(C2×C4), (C22×C4).362(C2×C4), SmallGroup(128,1718)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.304C23
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2b2, ab=ba, ac=ca, dad-1=a-1, eae-1=ab2, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2b2c, ede-1=a2d >
Subgroups: 204 in 165 conjugacy classes, 128 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C42.12C4, C42.7C22, C8×Q8, C8⋊4Q8, C23.37C23, C42.304C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C8○D4, C23×C4, 2- 1+4, C23.32C23, C2×C8○D4, C42.304C23
►On 64 points
(1 59 51 41)(2 60 52 42)(3 61 53 43)(4 62 54 44)(5 63 55 45)(6 64 56 46)(7 57 49 47)(8 58 50 48)(9 36 32 17)(10 37 25 18)(11 38 26 19)(12 39 27 20)(13 40 28 21)(14 33 29 22)(15 34 30 23)(16 35 31 24)
(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 12)(2 32 56 13)(3 25 49 14)(4 26 50 15)(5 27 51 16)(6 28 52 9)(7 29 53 10)(8 30 54 11)(17 64 40 42)(18 57 33 43)(19 58 34 44)(20 59 35 45)(21 60 36 46)(22 61 37 47)(23 62 38 48)(24 63 39 41)
(1 63 55 41)(2 42 56 64)(3 57 49 43)(4 44 50 58)(5 59 51 45)(6 46 52 60)(7 61 53 47)(8 48 54 62)(9 17 28 40)(10 33 29 18)(11 19 30 34)(12 35 31 20)(13 21 32 36)(14 37 25 22)(15 23 26 38)(16 39 27 24)
G:=sub<Sym(64)| (1,59,51,41)(2,60,52,42)(3,61,53,43)(4,62,54,44)(5,63,55,45)(6,64,56,46)(7,57,49,47)(8,58,50,48)(9,36,32,17)(10,37,25,18)(11,38,26,19)(12,39,27,20)(13,40,28,21)(14,33,29,22)(15,34,30,23)(16,35,31,24), (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), (1,31,55,12)(2,32,56,13)(3,25,49,14)(4,26,50,15)(5,27,51,16)(6,28,52,9)(7,29,53,10)(8,30,54,11)(17,64,40,42)(18,57,33,43)(19,58,34,44)(20,59,35,45)(21,60,36,46)(22,61,37,47)(23,62,38,48)(24,63,39,41), (1,63,55,41)(2,42,56,64)(3,57,49,43)(4,44,50,58)(5,59,51,45)(6,46,52,60)(7,61,53,47)(8,48,54,62)(9,17,28,40)(10,33,29,18)(11,19,30,34)(12,35,31,20)(13,21,32,36)(14,37,25,22)(15,23,26,38)(16,39,27,24)>;
G:=Group( (1,59,51,41)(2,60,52,42)(3,61,53,43)(4,62,54,44)(5,63,55,45)(6,64,56,46)(7,57,49,47)(8,58,50,48)(9,36,32,17)(10,37,25,18)(11,38,26,19)(12,39,27,20)(13,40,28,21)(14,33,29,22)(15,34,30,23)(16,35,31,24), (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), (1,31,55,12)(2,32,56,13)(3,25,49,14)(4,26,50,15)(5,27,51,16)(6,28,52,9)(7,29,53,10)(8,30,54,11)(17,64,40,42)(18,57,33,43)(19,58,34,44)(20,59,35,45)(21,60,36,46)(22,61,37,47)(23,62,38,48)(24,63,39,41), (1,63,55,41)(2,42,56,64)(3,57,49,43)(4,44,50,58)(5,59,51,45)(6,46,52,60)(7,61,53,47)(8,48,54,62)(9,17,28,40)(10,33,29,18)(11,19,30,34)(12,35,31,20)(13,21,32,36)(14,37,25,22)(15,23,26,38)(16,39,27,24) );
G=PermutationGroup([[(1,59,51,41),(2,60,52,42),(3,61,53,43),(4,62,54,44),(5,63,55,45),(6,64,56,46),(7,57,49,47),(8,58,50,48),(9,36,32,17),(10,37,25,18),(11,38,26,19),(12,39,27,20),(13,40,28,21),(14,33,29,22),(15,34,30,23),(16,35,31,24)], [(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)], [(1,31,55,12),(2,32,56,13),(3,25,49,14),(4,26,50,15),(5,27,51,16),(6,28,52,9),(7,29,53,10),(8,30,54,11),(17,64,40,42),(18,57,33,43),(19,58,34,44),(20,59,35,45),(21,60,36,46),(22,61,37,47),(23,62,38,48),(24,63,39,41)], [(1,63,55,41),(2,42,56,64),(3,57,49,43),(4,44,50,58),(5,59,51,45),(6,46,52,60),(7,61,53,47),(8,48,54,62),(9,17,28,40),(10,33,29,18),(11,19,30,34),(12,35,31,20),(13,21,32,36),(14,37,25,22),(15,23,26,38),(16,39,27,24)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4U | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 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 | 2 | 4 |
| type | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8○D4 | 2- 1+4 |
| kernel | C42.304C23 | C42.12C4 | C42.7C22 | C8×Q8 | C8⋊4Q8 | C23.37C23 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C4 | C4 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 8 | 4 | 4 | 16 | 2 |
Matrix representation of C42.304C23 ►in GL4(𝔽17) generated by
| 16 | 15 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 15 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 2 | 4 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 4 | 1 |
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 13 | 15 |
| 0 | 0 | 16 | 4 |
G:=sub<GL(4,GF(17))| [16,0,0,0,15,1,0,0,0,0,13,0,0,0,15,4],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[2,0,0,0,4,15,0,0,0,0,8,0,0,0,0,8],[4,0,0,0,0,4,0,0,0,0,16,4,0,0,0,1],[16,1,0,0,15,1,0,0,0,0,13,16,0,0,15,4] >;
C42.304C23 in GAP, Magma, Sage, TeX
C_4^2._{304}C_2^3 % in TeX
G:=Group("C4^2.304C2^3"); // GroupNames label
G:=SmallGroup(128,1718);
// by ID
G=gap.SmallGroup(128,1718);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,891,100,675,1018,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b,d^2=e^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,e*a*e^-1=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*b^2*c,e*d*e^-1=a^2*d>;
// generators/relations