p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.300C23, C4.1742+ (1+4), C4⋊Q8.33C4, C8⋊9D4⋊44C2, C4⋊1D4.20C4, C4⋊D4.28C4, C4⋊C8.236C22, (C2×C8).438C23, (C2×C4).677C24, C42.225(C2×C4), C4.4D4.21C4, (C4×D4).64C22, C8⋊C4.97C22, C42.6C4⋊52C2, C23.44(C22×C4), C22⋊C8.145C22, C2.31(Q8○M4(2)), (C22×C4).944C23, (C2×C42).784C22, C22.201(C23×C4), (C22×C8).450C22, C2.51(C22.11C24), (C2×M4(2)).247C22, C22.26C24.28C2, C4⋊C4.120(C2×C4), (C2×D4).144(C2×C4), C22⋊C4.21(C2×C4), (C2×Q8).126(C2×C4), (C22×C8)⋊C2⋊36C2, (C2×C4).277(C22×C4), (C22×C4).357(C2×C4), (C2×C4○D4).97C22, SmallGroup(128,1712)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 332 in 197 conjugacy classes, 124 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×10], Q8 [×2], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.6C4 [×2], C8⋊9D4 [×8], C22.26C24, C42.300C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ (1+4) [×2], C22.11C24, Q8○M4(2) [×2], C42.300C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a2c, ede=b2d >
►On 64 points
(1 18 55 33)(2 38 56 23)(3 20 49 35)(4 40 50 17)(5 22 51 37)(6 34 52 19)(7 24 53 39)(8 36 54 21)(9 58 28 44)(10 41 29 63)(11 60 30 46)(12 43 31 57)(13 62 32 48)(14 45 25 59)(15 64 26 42)(16 47 27 61)
(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 10)(2 30)(3 12)(4 32)(5 14)(6 26)(7 16)(8 28)(9 54)(11 56)(13 50)(15 52)(17 48)(18 63)(19 42)(20 57)(21 44)(22 59)(23 46)(24 61)(25 51)(27 53)(29 55)(31 49)(33 41)(34 64)(35 43)(36 58)(37 45)(38 60)(39 47)(40 62)
(1 12)(2 32)(3 14)(4 26)(5 16)(6 28)(7 10)(8 30)(9 52)(11 54)(13 56)(15 50)(17 64)(18 43)(19 58)(20 45)(21 60)(22 47)(23 62)(24 41)(25 49)(27 51)(29 53)(31 55)(33 57)(34 44)(35 59)(36 46)(37 61)(38 48)(39 63)(40 42)
G:=sub<Sym(64)| (1,18,55,33)(2,38,56,23)(3,20,49,35)(4,40,50,17)(5,22,51,37)(6,34,52,19)(7,24,53,39)(8,36,54,21)(9,58,28,44)(10,41,29,63)(11,60,30,46)(12,43,31,57)(13,62,32,48)(14,45,25,59)(15,64,26,42)(16,47,27,61), (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), (1,10)(2,30)(3,12)(4,32)(5,14)(6,26)(7,16)(8,28)(9,54)(11,56)(13,50)(15,52)(17,48)(18,63)(19,42)(20,57)(21,44)(22,59)(23,46)(24,61)(25,51)(27,53)(29,55)(31,49)(33,41)(34,64)(35,43)(36,58)(37,45)(38,60)(39,47)(40,62), (1,12)(2,32)(3,14)(4,26)(5,16)(6,28)(7,10)(8,30)(9,52)(11,54)(13,56)(15,50)(17,64)(18,43)(19,58)(20,45)(21,60)(22,47)(23,62)(24,41)(25,49)(27,51)(29,53)(31,55)(33,57)(34,44)(35,59)(36,46)(37,61)(38,48)(39,63)(40,42)>;
G:=Group( (1,18,55,33)(2,38,56,23)(3,20,49,35)(4,40,50,17)(5,22,51,37)(6,34,52,19)(7,24,53,39)(8,36,54,21)(9,58,28,44)(10,41,29,63)(11,60,30,46)(12,43,31,57)(13,62,32,48)(14,45,25,59)(15,64,26,42)(16,47,27,61), (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), (1,10)(2,30)(3,12)(4,32)(5,14)(6,26)(7,16)(8,28)(9,54)(11,56)(13,50)(15,52)(17,48)(18,63)(19,42)(20,57)(21,44)(22,59)(23,46)(24,61)(25,51)(27,53)(29,55)(31,49)(33,41)(34,64)(35,43)(36,58)(37,45)(38,60)(39,47)(40,62), (1,12)(2,32)(3,14)(4,26)(5,16)(6,28)(7,10)(8,30)(9,52)(11,54)(13,56)(15,50)(17,64)(18,43)(19,58)(20,45)(21,60)(22,47)(23,62)(24,41)(25,49)(27,51)(29,53)(31,55)(33,57)(34,44)(35,59)(36,46)(37,61)(38,48)(39,63)(40,42) );
G=PermutationGroup([(1,18,55,33),(2,38,56,23),(3,20,49,35),(4,40,50,17),(5,22,51,37),(6,34,52,19),(7,24,53,39),(8,36,54,21),(9,58,28,44),(10,41,29,63),(11,60,30,46),(12,43,31,57),(13,62,32,48),(14,45,25,59),(15,64,26,42),(16,47,27,61)], [(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)], [(1,10),(2,30),(3,12),(4,32),(5,14),(6,26),(7,16),(8,28),(9,54),(11,56),(13,50),(15,52),(17,48),(18,63),(19,42),(20,57),(21,44),(22,59),(23,46),(24,61),(25,51),(27,53),(29,55),(31,49),(33,41),(34,64),(35,43),(36,58),(37,45),(38,60),(39,47),(40,62)], [(1,12),(2,32),(3,14),(4,26),(5,16),(6,28),(7,10),(8,30),(9,52),(11,54),(13,56),(15,50),(17,64),(18,43),(19,58),(20,45),(21,60),(22,47),(23,62),(24,41),(25,49),(27,51),(29,53),(31,55),(33,57),(34,44),(35,59),(36,46),(37,61),(38,48),(39,63),(40,42)])
Matrix representation ►G ⊆ GL8(𝔽17)
| 13 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 4 | 0 | 13 | 4 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 15 | 0 | 1 |
| 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 6 | 6 | 0 | 0 | 0 | 0 |
| 16 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 11 | 11 | 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 4 | 4 |
| 0 | 0 | 0 | 0 | 8 | 0 | 9 | 13 |
| 4 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 1 | 1 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 16 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 1 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 16 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(17))| [13,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,9,4,13,4,0,0,0,0,0,0,0,0,16,0,0,15,0,0,0,0,16,0,1,15,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,1],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,7,16,11,0,0,0,0,0,7,0,11,0,0,0,0,5,6,1,6,0,0,0,0,0,6,0,10,0,0,0,0,0,0,0,0,0,4,13,8,0,0,0,0,13,0,13,0,0,0,0,0,0,0,4,9,0,0,0,0,0,0,4,13],[4,13,4,0,0,0,0,0,8,13,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,16,2,0,0,0,0,1,0,16,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16],[16,0,16,1,0,0,0,0,15,1,16,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4M | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | 2+ (1+4) | Q8○M4(2) |
| kernel | C42.300C23 | (C22×C8)⋊C2 | C42.6C4 | C8⋊9D4 | C22.26C24 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | C4 | C2 |
| # reps | 1 | 4 | 2 | 8 | 1 | 8 | 4 | 2 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{300}C_2^3 % in TeX
G:=Group("C4^2.300C2^3"); // GroupNames label
G:=SmallGroup(128,1712);
// by ID
G=gap.SmallGroup(128,1712);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,1018,521,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*c,e*d*e=b^2*d>;
// generators/relations