p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.305C23, C4.1222- 1+4, C4⋊Q8.37C4, C8⋊4Q8⋊43C2, C22⋊Q8.29C4, C4⋊C8.239C22, (C2×C4).684C24, C42.232(C2×C4), (C4×C8).342C22, (C2×C8).444C23, C42.C2.22C4, (C4×Q8).61C22, C8⋊C4.103C22, C22⋊C8.147C22, C2.35(Q8○M4(2)), C42.6C4.35C2, (C2×C42).791C22, C22.207(C23×C4), (C22×C4).948C23, C23.109(C22×C4), C42⋊C2.90C22, C42.7C22.5C2, C23.37C23.26C2, C2.27(C23.32C23), C4⋊C4.123(C2×C4), C22⋊C4.25(C2×C4), (C2×C4).86(C22×C4), (C2×Q8).128(C2×C4), (C22×C4).363(C2×C4), SmallGroup(128,1719)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.305C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, eae=ab2, bc=cb, bd=db, be=eb, dcd-1=b2c, ece=a2c, de=ed >
Subgroups: 204 in 157 conjugacy classes, 124 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, C42.6C4, C42.7C22, C8⋊4Q8, C23.37C23, C42.305C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2- 1+4, C23.32C23, Q8○M4(2), C42.305C23
►On 64 points
Generators in S64
(1 63 55 16)(2 9 56 64)(3 57 49 10)(4 11 50 58)(5 59 51 12)(6 13 52 60)(7 61 53 14)(8 15 54 62)(17 33 29 47)(18 48 30 34)(19 35 31 41)(20 42 32 36)(21 37 25 43)(22 44 26 38)(23 39 27 45)(24 46 28 40)
(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 19)(2 28 56 24)(3 25 49 21)(4 30 50 18)(5 27 51 23)(6 32 52 20)(7 29 53 17)(8 26 54 22)(9 42 64 36)(10 47 57 33)(11 44 58 38)(12 41 59 35)(13 46 60 40)(14 43 61 37)(15 48 62 34)(16 45 63 39)
(2 56)(4 50)(6 52)(8 54)(9 60)(10 14)(11 62)(12 16)(13 64)(15 58)(18 30)(20 32)(22 26)(24 28)(33 37)(34 44)(35 39)(36 46)(38 48)(40 42)(41 45)(43 47)(57 61)(59 63)
G:=sub<Sym(64)| (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,33,29,47)(18,48,30,34)(19,35,31,41)(20,42,32,36)(21,37,25,43)(22,44,26,38)(23,39,27,45)(24,46,28,40), (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), (1,31,55,19)(2,28,56,24)(3,25,49,21)(4,30,50,18)(5,27,51,23)(6,32,52,20)(7,29,53,17)(8,26,54,22)(9,42,64,36)(10,47,57,33)(11,44,58,38)(12,41,59,35)(13,46,60,40)(14,43,61,37)(15,48,62,34)(16,45,63,39), (2,56)(4,50)(6,52)(8,54)(9,60)(10,14)(11,62)(12,16)(13,64)(15,58)(18,30)(20,32)(22,26)(24,28)(33,37)(34,44)(35,39)(36,46)(38,48)(40,42)(41,45)(43,47)(57,61)(59,63)>;
G:=Group( (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,33,29,47)(18,48,30,34)(19,35,31,41)(20,42,32,36)(21,37,25,43)(22,44,26,38)(23,39,27,45)(24,46,28,40), (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), (1,31,55,19)(2,28,56,24)(3,25,49,21)(4,30,50,18)(5,27,51,23)(6,32,52,20)(7,29,53,17)(8,26,54,22)(9,42,64,36)(10,47,57,33)(11,44,58,38)(12,41,59,35)(13,46,60,40)(14,43,61,37)(15,48,62,34)(16,45,63,39), (2,56)(4,50)(6,52)(8,54)(9,60)(10,14)(11,62)(12,16)(13,64)(15,58)(18,30)(20,32)(22,26)(24,28)(33,37)(34,44)(35,39)(36,46)(38,48)(40,42)(41,45)(43,47)(57,61)(59,63) );
G=PermutationGroup([[(1,63,55,16),(2,9,56,64),(3,57,49,10),(4,11,50,58),(5,59,51,12),(6,13,52,60),(7,61,53,14),(8,15,54,62),(17,33,29,47),(18,48,30,34),(19,35,31,41),(20,42,32,36),(21,37,25,43),(22,44,26,38),(23,39,27,45),(24,46,28,40)], [(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)], [(1,31,55,19),(2,28,56,24),(3,25,49,21),(4,30,50,18),(5,27,51,23),(6,32,52,20),(7,29,53,17),(8,26,54,22),(9,42,64,36),(10,47,57,33),(11,44,58,38),(12,41,59,35),(13,46,60,40),(14,43,61,37),(15,48,62,34),(16,45,63,39)], [(2,56),(4,50),(6,52),(8,54),(9,60),(10,14),(11,62),(12,16),(13,64),(15,58),(18,30),(20,32),(22,26),(24,28),(33,37),(34,44),(35,39),(36,46),(38,48),(40,42),(41,45),(43,47),(57,61),(59,63)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4Q | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | 2- 1+4 | Q8○M4(2) |
| kernel | C42.305C23 | C42.6C4 | C42.7C22 | C8⋊4Q8 | C23.37C23 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C4 | C2 |
| # reps | 1 | 2 | 4 | 8 | 1 | 8 | 4 | 4 | 2 | 4 |
Matrix representation of C42.305C23 ►in GL8(𝔽17)
| 1 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 13 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 9 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 9 | 0 | 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 9 | 8 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 12 | 15 |
| 0 | 0 | 0 | 0 | 14 | 8 | 2 | 5 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 16 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 13 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [1,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,13,16,13,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[9,9,0,9,0,0,0,0,0,0,0,8,0,0,0,0,16,8,8,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,4,14,0,14,0,0,0,0,4,13,2,8,0,0,0,0,0,0,12,2,0,0,0,0,0,0,15,5],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,16,4,0,0,0,0,0,15,13,16,13,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
C42.305C23 in GAP, Magma, Sage, TeX
C_4^2._{305}C_2^3 % in TeX
G:=Group("C4^2.305C2^3"); // GroupNames label
G:=SmallGroup(128,1719);
// by ID
G=gap.SmallGroup(128,1719);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,891,100,675,1018,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e=a^2*c,d*e=e*d>;
// generators/relations