p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.353C23, (C4×D8)⋊23C2, C4⋊D8⋊23C2, C4⋊C4.346D4, C4⋊SD16⋊7C2, (C4×SD16)⋊6C2, Q8.Q8⋊19C2, D4.Q8⋊19C2, D8⋊C4⋊10C2, D4⋊D4⋊20C2, C2.18(D4○D8), C4⋊C8.54C22, C4⋊C4.72C23, (C2×C8).46C23, Q8.9(C4○D4), D4.10(C4○D4), (C2×C4).317C24, (C4×C8).110C22, C22⋊C4.147D4, (C4×D4).80C22, C23.256(C2×D4), SD16⋊C4⋊13C2, (C4×Q8).77C22, C8⋊C4.11C22, C4.Q8.20C22, C2.27(D4○SD16), (C2×D8).126C22, (C2×D4).408C23, C22.D8⋊17C2, C4⋊1D4.60C22, C4⋊D4.27C22, C23.46D4⋊6C2, C22⋊C8.30C22, (C2×Q8).380C23, C2.D8.174C22, D4⋊C4.35C22, (C22×C4).290C23, C42.7C22⋊2C2, (C2×SD16).18C22, C22.577(C22×D4), C42.C2.12C22, C22.34C24⋊1C2, Q8⋊C4.155C22, C23.33C23⋊12C2, C42⋊C2.128C22, C2.118(C22.19C24), C4.202(C2×C4○D4), (C2×C4).501(C2×D4), (C2×C4⋊C4).614C22, (C2×C4○D4).143C22, SmallGroup(128,1851)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 404 in 198 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×13], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×2], D4 [×13], Q8 [×2], Q8, C23, C23 [×3], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×5], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×6], C2×D4 [×3], C2×D4 [×5], C2×Q8, C4○D4 [×4], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×4], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22.D4 [×2], C42.C2, C4⋊1D4, C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C42.7C22, C4×D8, C4×SD16, SD16⋊C4, D8⋊C4, D4⋊D4 [×2], C4⋊D8, C4⋊SD16, D4.Q8, Q8.Q8, C22.D8, C23.46D4, C23.33C23, C22.34C24, C42.353C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○D8, D4○SD16, C42.353C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, 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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 63 10)(6 16 64 11)(7 13 61 12)(8 14 62 9)(29 36 41 38)(30 33 42 39)(31 34 43 40)(32 35 44 37)(45 55 58 50)(46 56 59 51)(47 53 60 52)(48 54 57 49)
(5 13)(6 14)(7 15)(8 16)(9 64)(10 61)(11 62)(12 63)(17 23)(18 24)(19 21)(20 22)(29 36)(30 33)(31 34)(32 35)(37 44)(38 41)(39 42)(40 43)(45 47)(46 48)(49 56)(50 53)(51 54)(52 55)(57 59)(58 60)
(1 48 25 57)(2 58 26 45)(3 46 27 59)(4 60 28 47)(5 44 63 32)(6 29 64 41)(7 42 61 30)(8 31 62 43)(9 40 14 34)(10 35 15 37)(11 38 16 36)(12 33 13 39)(17 49 23 54)(18 55 24 50)(19 51 21 56)(20 53 22 52)
(1 29)(2 30)(3 31)(4 32)(5 60)(6 57)(7 58)(8 59)(9 51)(10 52)(11 49)(12 50)(13 55)(14 56)(15 53)(16 54)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(25 41)(26 42)(27 43)(28 44)(45 61)(46 62)(47 63)(48 64)
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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,63,10)(6,16,64,11)(7,13,61,12)(8,14,62,9)(29,36,41,38)(30,33,42,39)(31,34,43,40)(32,35,44,37)(45,55,58,50)(46,56,59,51)(47,53,60,52)(48,54,57,49), (5,13)(6,14)(7,15)(8,16)(9,64)(10,61)(11,62)(12,63)(17,23)(18,24)(19,21)(20,22)(29,36)(30,33)(31,34)(32,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48)(49,56)(50,53)(51,54)(52,55)(57,59)(58,60), (1,48,25,57)(2,58,26,45)(3,46,27,59)(4,60,28,47)(5,44,63,32)(6,29,64,41)(7,42,61,30)(8,31,62,43)(9,40,14,34)(10,35,15,37)(11,38,16,36)(12,33,13,39)(17,49,23,54)(18,55,24,50)(19,51,21,56)(20,53,22,52), (1,29)(2,30)(3,31)(4,32)(5,60)(6,57)(7,58)(8,59)(9,51)(10,52)(11,49)(12,50)(13,55)(14,56)(15,53)(16,54)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64)>;
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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,63,10)(6,16,64,11)(7,13,61,12)(8,14,62,9)(29,36,41,38)(30,33,42,39)(31,34,43,40)(32,35,44,37)(45,55,58,50)(46,56,59,51)(47,53,60,52)(48,54,57,49), (5,13)(6,14)(7,15)(8,16)(9,64)(10,61)(11,62)(12,63)(17,23)(18,24)(19,21)(20,22)(29,36)(30,33)(31,34)(32,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48)(49,56)(50,53)(51,54)(52,55)(57,59)(58,60), (1,48,25,57)(2,58,26,45)(3,46,27,59)(4,60,28,47)(5,44,63,32)(6,29,64,41)(7,42,61,30)(8,31,62,43)(9,40,14,34)(10,35,15,37)(11,38,16,36)(12,33,13,39)(17,49,23,54)(18,55,24,50)(19,51,21,56)(20,53,22,52), (1,29)(2,30)(3,31)(4,32)(5,60)(6,57)(7,58)(8,59)(9,51)(10,52)(11,49)(12,50)(13,55)(14,56)(15,53)(16,54)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64) );
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)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,63,10),(6,16,64,11),(7,13,61,12),(8,14,62,9),(29,36,41,38),(30,33,42,39),(31,34,43,40),(32,35,44,37),(45,55,58,50),(46,56,59,51),(47,53,60,52),(48,54,57,49)], [(5,13),(6,14),(7,15),(8,16),(9,64),(10,61),(11,62),(12,63),(17,23),(18,24),(19,21),(20,22),(29,36),(30,33),(31,34),(32,35),(37,44),(38,41),(39,42),(40,43),(45,47),(46,48),(49,56),(50,53),(51,54),(52,55),(57,59),(58,60)], [(1,48,25,57),(2,58,26,45),(3,46,27,59),(4,60,28,47),(5,44,63,32),(6,29,64,41),(7,42,61,30),(8,31,62,43),(9,40,14,34),(10,35,15,37),(11,38,16,36),(12,33,13,39),(17,49,23,54),(18,55,24,50),(19,51,21,56),(20,53,22,52)], [(1,29),(2,30),(3,31),(4,32),(5,60),(6,57),(7,58),(8,59),(9,51),(10,52),(11,49),(12,50),(13,55),(14,56),(15,53),(16,54),(17,36),(18,33),(19,34),(20,35),(21,40),(22,37),(23,38),(24,39),(25,41),(26,42),(27,43),(28,44),(45,61),(46,62),(47,63),(48,64)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 16 | 13 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 1 |
| 0 | 0 | 4 | 13 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 0 | 0 | 16 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 11 | 0 | 6 |
| 0 | 0 | 8 | 14 | 3 | 3 |
| 0 | 0 | 15 | 2 | 0 | 9 |
| 0 | 0 | 0 | 5 | 3 | 12 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 10 | 10 |
| 0 | 0 | 14 | 0 | 0 | 10 |
| 0 | 0 | 0 | 5 | 3 | 3 |
| 0 | 0 | 5 | 12 | 3 | 3 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,15,16,13,13,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,4,0,0,15,16,13,13,0,0,0,0,0,16,0,0,0,0,1,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,4,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,8,8,15,0,0,0,11,14,2,5,0,0,0,3,0,3,0,0,6,3,9,12],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,11,14,0,5,0,0,0,0,5,12,0,0,10,0,3,3,0,0,10,10,3,3] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | ··· | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | D4○D8 | D4○SD16 |
| kernel | C42.353C23 | C42.7C22 | C4×D8 | C4×SD16 | SD16⋊C4 | D8⋊C4 | D4⋊D4 | C4⋊D8 | C4⋊SD16 | D4.Q8 | Q8.Q8 | C22.D8 | C23.46D4 | C23.33C23 | C22.34C24 | C22⋊C4 | C4⋊C4 | D4 | Q8 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{353}C_2^3 % in TeX
G:=Group("C4^2.353C2^3"); // GroupNames label
G:=SmallGroup(128,1851);
// by ID
G=gap.SmallGroup(128,1851);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,1018,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e=b*c,d*e=e*d>;
// generators/relations