p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.465C23, C4.452+ (1+4), (C8×D4)⋊18C2, Q8.Q8⋊5C2, C8⋊8D4⋊38C2, C4⋊C4.264D4, (C4×Q16)⋊12C2, (C2×D4).235D4, (C4×C8).84C22, Q8⋊5D4.2C2, Q8.D4⋊8C2, C2.45(Q8○D8), D4.7D4⋊10C2, C8.18D4⋊13C2, C4⋊C4.229C23, C4⋊C8.318C22, (C2×C8).184C23, (C2×C4).492C24, Q8.17(C4○D4), C22⋊Q16⋊10C2, C22⋊C4.104D4, C22.D8⋊8C2, C23.108(C2×D4), C2.D8.54C22, (C4×D4).333C22, (C2×D4).223C23, C22.12(C4○D8), C23.20D4⋊4C2, C4⋊D4.73C22, (C2×Q8).209C23, (C4×Q8).149C22, C2.128(D4⋊5D4), C4.Q8.100C22, C22⋊Q8.72C22, D4⋊C4.12C22, C23.24D4⋊11C2, C22⋊C8.201C22, (C22×C8).161C22, (C2×Q16).133C22, Q8⋊C4.12C22, (C2×SD16).96C22, C4.4D4.60C22, C22.752(C22×D4), C42.C2.35C22, C22.46C24⋊2C2, C42.78C22⋊5C2, (C22×C4).1136C23, (C22×Q8).339C22, C42⋊C2.180C22, C2.60(C2×C4○D8), C4.217(C2×C4○D4), (C2×C4).169(C2×D4), (C2×Q8⋊C4)⋊42C2, C22⋊C4○(Q8⋊C4), (C2×C4⋊C4).662C22, (C2×C4○D4).198C22, SmallGroup(128,2032)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 360 in 193 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×5], C2×C4 [×17], D4 [×8], Q8 [×2], Q8 [×7], C23 [×2], C23, C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×5], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×2], SD16, Q16 [×3], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×3], C4×C8, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×7], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C22×C8 [×2], C2×SD16, C2×Q16 [×2], C22×Q8, C2×C4○D4, C2×Q8⋊C4, C23.24D4, C8×D4, C4×Q16, C22⋊Q16, D4.7D4, Q8.D4, C8⋊8D4, C8.18D4, Q8.Q8, C22.D8, C23.20D4, C42.78C22, Q8⋊5D4, C22.46C24, C42.465C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2+ (1+4), D4⋊5D4, C2×C4○D8, Q8○D8, C42.465C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
►On 64 points
(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 41 18 46)(2 42 19 47)(3 43 20 48)(4 44 17 45)(5 36 61 37)(6 33 62 38)(7 34 63 39)(8 35 64 40)(9 21 13 50)(10 22 14 51)(11 23 15 52)(12 24 16 49)(25 57 32 54)(26 58 29 55)(27 59 30 56)(28 60 31 53)
(1 58 20 53)(2 54 17 59)(3 60 18 55)(4 56 19 57)(5 23 63 50)(6 51 64 24)(7 21 61 52)(8 49 62 22)(9 37 15 34)(10 35 16 38)(11 39 13 36)(12 33 14 40)(25 44 30 47)(26 48 31 41)(27 42 32 45)(28 46 29 43)
(1 12 18 16)(2 9 19 13)(3 10 20 14)(4 11 17 15)(5 54 61 57)(6 55 62 58)(7 56 63 59)(8 53 64 60)(21 42 50 47)(22 43 51 48)(23 44 52 45)(24 41 49 46)(25 36 32 37)(26 33 29 38)(27 34 30 39)(28 35 31 40)
(1 16 18 12)(2 11 19 15)(3 14 20 10)(4 9 17 13)(5 25 61 32)(6 31 62 28)(7 27 63 30)(8 29 64 26)(21 45 50 44)(22 43 51 48)(23 47 52 42)(24 41 49 46)(33 53 38 60)(34 59 39 56)(35 55 40 58)(36 57 37 54)
G:=sub<Sym(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), (1,41,18,46)(2,42,19,47)(3,43,20,48)(4,44,17,45)(5,36,61,37)(6,33,62,38)(7,34,63,39)(8,35,64,40)(9,21,13,50)(10,22,14,51)(11,23,15,52)(12,24,16,49)(25,57,32,54)(26,58,29,55)(27,59,30,56)(28,60,31,53), (1,58,20,53)(2,54,17,59)(3,60,18,55)(4,56,19,57)(5,23,63,50)(6,51,64,24)(7,21,61,52)(8,49,62,22)(9,37,15,34)(10,35,16,38)(11,39,13,36)(12,33,14,40)(25,44,30,47)(26,48,31,41)(27,42,32,45)(28,46,29,43), (1,12,18,16)(2,9,19,13)(3,10,20,14)(4,11,17,15)(5,54,61,57)(6,55,62,58)(7,56,63,59)(8,53,64,60)(21,42,50,47)(22,43,51,48)(23,44,52,45)(24,41,49,46)(25,36,32,37)(26,33,29,38)(27,34,30,39)(28,35,31,40), (1,16,18,12)(2,11,19,15)(3,14,20,10)(4,9,17,13)(5,25,61,32)(6,31,62,28)(7,27,63,30)(8,29,64,26)(21,45,50,44)(22,43,51,48)(23,47,52,42)(24,41,49,46)(33,53,38,60)(34,59,39,56)(35,55,40,58)(36,57,37,54)>;
G:=Group( (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), (1,41,18,46)(2,42,19,47)(3,43,20,48)(4,44,17,45)(5,36,61,37)(6,33,62,38)(7,34,63,39)(8,35,64,40)(9,21,13,50)(10,22,14,51)(11,23,15,52)(12,24,16,49)(25,57,32,54)(26,58,29,55)(27,59,30,56)(28,60,31,53), (1,58,20,53)(2,54,17,59)(3,60,18,55)(4,56,19,57)(5,23,63,50)(6,51,64,24)(7,21,61,52)(8,49,62,22)(9,37,15,34)(10,35,16,38)(11,39,13,36)(12,33,14,40)(25,44,30,47)(26,48,31,41)(27,42,32,45)(28,46,29,43), (1,12,18,16)(2,9,19,13)(3,10,20,14)(4,11,17,15)(5,54,61,57)(6,55,62,58)(7,56,63,59)(8,53,64,60)(21,42,50,47)(22,43,51,48)(23,44,52,45)(24,41,49,46)(25,36,32,37)(26,33,29,38)(27,34,30,39)(28,35,31,40), (1,16,18,12)(2,11,19,15)(3,14,20,10)(4,9,17,13)(5,25,61,32)(6,31,62,28)(7,27,63,30)(8,29,64,26)(21,45,50,44)(22,43,51,48)(23,47,52,42)(24,41,49,46)(33,53,38,60)(34,59,39,56)(35,55,40,58)(36,57,37,54) );
G=PermutationGroup([(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)], [(1,41,18,46),(2,42,19,47),(3,43,20,48),(4,44,17,45),(5,36,61,37),(6,33,62,38),(7,34,63,39),(8,35,64,40),(9,21,13,50),(10,22,14,51),(11,23,15,52),(12,24,16,49),(25,57,32,54),(26,58,29,55),(27,59,30,56),(28,60,31,53)], [(1,58,20,53),(2,54,17,59),(3,60,18,55),(4,56,19,57),(5,23,63,50),(6,51,64,24),(7,21,61,52),(8,49,62,22),(9,37,15,34),(10,35,16,38),(11,39,13,36),(12,33,14,40),(25,44,30,47),(26,48,31,41),(27,42,32,45),(28,46,29,43)], [(1,12,18,16),(2,9,19,13),(3,10,20,14),(4,11,17,15),(5,54,61,57),(6,55,62,58),(7,56,63,59),(8,53,64,60),(21,42,50,47),(22,43,51,48),(23,44,52,45),(24,41,49,46),(25,36,32,37),(26,33,29,38),(27,34,30,39),(28,35,31,40)], [(1,16,18,12),(2,11,19,15),(3,14,20,10),(4,9,17,13),(5,25,61,32),(6,31,62,28),(7,27,63,30),(8,29,64,26),(21,45,50,44),(22,43,51,48),(23,47,52,42),(24,41,49,46),(33,53,38,60),(34,59,39,56),(35,55,40,58),(36,57,37,54)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 13 |
| 0 | 0 | 9 | 16 |
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 11 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 16 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,1,9,0,0,13,16],[16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,11,6,0,0,0,0,4,0,0,0,1,13],[13,4,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,9,0,0,0,16] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 4M | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2+ (1+4) | Q8○D8 |
| kernel | C42.465C23 | C2×Q8⋊C4 | C23.24D4 | C8×D4 | C4×Q16 | C22⋊Q16 | D4.7D4 | Q8.D4 | C8⋊8D4 | C8.18D4 | Q8.Q8 | C22.D8 | C23.20D4 | C42.78C22 | Q8⋊5D4 | C22.46C24 | C22⋊C4 | C4⋊C4 | C2×D4 | Q8 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{465}C_2^3 % in TeX
G:=Group("C4^2.465C2^3"); // GroupNames label
G:=SmallGroup(128,2032);
// by ID
G=gap.SmallGroup(128,2032);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,352,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2*b^2,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations