p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.29C42, (C22×C8)⋊10C4, (C23×C8).2C2, (C2×C4).63C42, C24.92(C2×C4), (C22×C4).84Q8, C23.56(C4⋊C4), (C2×M4(2))⋊10C4, (C22×C4).750D4, (C2×C42).1C22, C22.19(C8○D4), C22.46(C2×C42), C23.246(C22×C4), (C23×C4).665C22, (C22×C8).371C22, (C22×M4(2)).7C2, C2.10(C8○2M4(2)), C23.113(C22⋊C4), C4.12(C2.C42), C22.7C42⋊34C2, (C22×C4).1599C23, C22.8(C2.C42), C2.1(C42.6C22), C4.71(C2×C4⋊C4), (C2×C4⋊C4).44C4, (C2×C8).126(C2×C4), C22.51(C2×C4⋊C4), (C2×C4).328(C2×Q8), (C2×C4).118(C4⋊C4), (C2×C4).1490(C2×D4), (C2×C22⋊C4).19C4, C4.101(C2×C22⋊C4), (C2×C42⋊C2).3C2, (C22×C4).104(C2×C4), (C2×C4).589(C22×C4), C2.6(C2×C2.C42), (C2×C4).250(C22⋊C4), C2.1((C22×C8)⋊C2), C22.102(C2×C22⋊C4), SmallGroup(128,461)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.29C42
G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >
Subgroups: 308 in 208 conjugacy classes, 116 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22.7C42, C2×C42⋊C2, C23×C8, C22×M4(2), C23.29C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C8○D4, C2×C2.C42, C8○2M4(2), (C22×C8)⋊C2, C42.6C22, C23.29C42
►On 64 points
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 63 55 16)(2 38 56 46)(3 57 49 10)(4 40 50 48)(5 59 51 12)(6 34 52 42)(7 61 53 14)(8 36 54 44)(9 18 64 32)(11 20 58 26)(13 22 60 28)(15 24 62 30)(17 37 31 45)(19 39 25 47)(21 33 27 41)(23 35 29 43)
G:=sub<Sym(64)| (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,63,55,16)(2,38,56,46)(3,57,49,10)(4,40,50,48)(5,59,51,12)(6,34,52,42)(7,61,53,14)(8,36,54,44)(9,18,64,32)(11,20,58,26)(13,22,60,28)(15,24,62,30)(17,37,31,45)(19,39,25,47)(21,33,27,41)(23,35,29,43)>;
G:=Group( (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,63,55,16)(2,38,56,46)(3,57,49,10)(4,40,50,48)(5,59,51,12)(6,34,52,42)(7,61,53,14)(8,36,54,44)(9,18,64,32)(11,20,58,26)(13,22,60,28)(15,24,62,30)(17,37,31,45)(19,39,25,47)(21,33,27,41)(23,35,29,43) );
G=PermutationGroup([[(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,57),(40,58)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,63,55,16),(2,38,56,46),(3,57,49,10),(4,40,50,48),(5,59,51,12),(6,34,52,42),(7,61,53,14),(8,36,54,44),(9,18,64,32),(11,20,58,26),(13,22,60,28),(15,24,62,30),(17,37,31,45),(19,39,25,47),(21,33,27,41),(23,35,29,43)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | C8○D4 |
| kernel | C23.29C42 | C22.7C42 | C2×C42⋊C2 | C23×C8 | C22×M4(2) | C2×C22⋊C4 | C2×C4⋊C4 | C22×C8 | C2×M4(2) | C22×C4 | C22×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 6 | 2 | 16 |
Matrix representation of C23.29C42 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 13 | 15 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 16 | 4 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,8,0,0,0,0,0,9,0,0,0,0,0,13,0,0,0,0,15,4],[4,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,13,16,0,0,0,0,4] >;
C23.29C42 in GAP, Magma, Sage, TeX
C_2^3._{29}C_4^2 % in TeX
G:=Group("C2^3.29C4^2"); // GroupNames label
G:=SmallGroup(128,461);
// by ID
G=gap.SmallGroup(128,461);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^4=1,d^4=c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c>;
// generators/relations