p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.697C23, C4.1762+ (1+4), C4.1232- (1+4), D4○2(C4⋊C8), C4○D4⋊3C8, D4⋊8(C2×C8), (C8×D4)⋊4C2, Q8○2(C4⋊C8), (C8×Q8)⋊4C2, Q8⋊7(C2×C8), (C4×D4).36C4, C2.9(C23×C8), (C4×Q8).33C4, (C4×C8).32C22, C4.21(C22×C8), C4⋊C8.377C22, (C2×C8).483C23, (C2×C4).685C24, C42.233(C2×C4), C22.3(C22×C8), C42⋊C2.34C4, (C4×D4).364C22, C2.5(Q8○M4(2)), (C22×C8).95C22, C22.45(C23×C4), (C4×Q8).335C22, C42.12C4⋊24C2, C22⋊C8.246C22, C23.151(C22×C4), (C2×C42).792C22, (C22×C4).1286C23, C2.4(C23.33C23), C4⋊C8○(C4×D4), C4⋊C4○(C4⋊C8), C4⋊C8○(C4×Q8), (C2×C4)⋊5(C2×C8), (C2×D4)○(C4⋊C8), (C2×C4⋊C8)⋊19C2, C22⋊C4○(C4⋊C8), C22⋊C8○(C4⋊C8), C4⋊C4.252(C2×C4), (C4×C4○D4).18C2, (C2×C4○D4).28C4, (C2×D4).253(C2×C4), C22⋊C4.94(C2×C4), (C2×Q8).230(C2×C4), (C2×C4).500(C22×C4), (C22×C4).141(C2×C4), SmallGroup(128,1720)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 276 in 216 conjugacy classes, 174 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×8], C4 [×7], C22, C22 [×6], C22 [×6], C8 [×8], C2×C4 [×3], C2×C4 [×21], C2×C4 [×9], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×6], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×C8 [×6], C22⋊C8 [×6], C4⋊C8, C4⋊C8 [×9], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×C4○D4, C2×C4⋊C8 [×3], C42.12C4 [×3], C8×D4 [×6], C8×Q8 [×2], C4×C4○D4, C42.697C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2+ (1+4), 2- (1+4), C23.33C23, C23×C8, Q8○M4(2), C42.697C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, ede=a2d >
►On 64 points
(1 16 55 63)(2 9 56 64)(3 10 49 57)(4 11 50 58)(5 12 51 59)(6 13 52 60)(7 14 53 61)(8 15 54 62)(17 45 31 35)(18 46 32 36)(19 47 25 37)(20 48 26 38)(21 41 27 39)(22 42 28 40)(23 43 29 33)(24 44 30 34)
(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 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 45)(2 36)(3 47)(4 38)(5 41)(6 40)(7 43)(8 34)(9 18)(10 25)(11 20)(12 27)(13 22)(14 29)(15 24)(16 31)(17 63)(19 57)(21 59)(23 61)(26 58)(28 60)(30 62)(32 64)(33 53)(35 55)(37 49)(39 51)(42 52)(44 54)(46 56)(48 50)
G:=sub<Sym(64)| (1,16,55,63)(2,9,56,64)(3,10,49,57)(4,11,50,58)(5,12,51,59)(6,13,52,60)(7,14,53,61)(8,15,54,62)(17,45,31,35)(18,46,32,36)(19,47,25,37)(20,48,26,38)(21,41,27,39)(22,42,28,40)(23,43,29,33)(24,44,30,34), (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), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,45)(2,36)(3,47)(4,38)(5,41)(6,40)(7,43)(8,34)(9,18)(10,25)(11,20)(12,27)(13,22)(14,29)(15,24)(16,31)(17,63)(19,57)(21,59)(23,61)(26,58)(28,60)(30,62)(32,64)(33,53)(35,55)(37,49)(39,51)(42,52)(44,54)(46,56)(48,50)>;
G:=Group( (1,16,55,63)(2,9,56,64)(3,10,49,57)(4,11,50,58)(5,12,51,59)(6,13,52,60)(7,14,53,61)(8,15,54,62)(17,45,31,35)(18,46,32,36)(19,47,25,37)(20,48,26,38)(21,41,27,39)(22,42,28,40)(23,43,29,33)(24,44,30,34), (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), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,45)(2,36)(3,47)(4,38)(5,41)(6,40)(7,43)(8,34)(9,18)(10,25)(11,20)(12,27)(13,22)(14,29)(15,24)(16,31)(17,63)(19,57)(21,59)(23,61)(26,58)(28,60)(30,62)(32,64)(33,53)(35,55)(37,49)(39,51)(42,52)(44,54)(46,56)(48,50) );
G=PermutationGroup([(1,16,55,63),(2,9,56,64),(3,10,49,57),(4,11,50,58),(5,12,51,59),(6,13,52,60),(7,14,53,61),(8,15,54,62),(17,45,31,35),(18,46,32,36),(19,47,25,37),(20,48,26,38),(21,41,27,39),(22,42,28,40),(23,43,29,33),(24,44,30,34)], [(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)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,45),(18,46),(19,47),(20,48),(21,41),(22,42),(23,43),(24,44),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(1,45),(2,36),(3,47),(4,38),(5,41),(6,40),(7,43),(8,34),(9,18),(10,25),(11,20),(12,27),(13,22),(14,29),(15,24),(16,31),(17,63),(19,57),(21,59),(23,61),(26,58),(28,60),(30,62),(32,64),(33,53),(35,55),(37,49),(39,51),(42,52),(44,54),(46,56),(48,50)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 7 | 1 | 0 | 0 |
| 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 1 | 10 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,16,0,0,0,0,0,16,0,0],[13,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[9,0,0,0,0,0,7,1,0,0,0,1,10,0,0,0,0,0,7,1,0,0,0,1,10],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,16,0,0,0,16,0,0,0,1,0,0,0] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4Z | 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 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | 2+ (1+4) | 2- (1+4) | Q8○M4(2) |
| kernel | C42.697C23 | C2×C4⋊C8 | C42.12C4 | C8×D4 | C8×Q8 | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C4 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 6 | 6 | 2 | 2 | 32 | 1 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{697}C_2^3 % in TeX
G:=Group("C4^2.697C2^3"); // GroupNames label
G:=SmallGroup(128,1720);
// by ID
G=gap.SmallGroup(128,1720);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,a*c=c*a,d*a*d=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=a^2*c,e*d*e=a^2*d>;
// generators/relations