p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.261C23, (C4×D4).24C4, (C4×Q8).23C4, C4⋊C8.230C22, (C4×M4(2))⋊32C2, (C2×C8).399C23, C42.205(C2×C4), (C4×C8).326C22, (C2×C4).642C24, C42.6C4⋊44C2, C8⋊C4.153C22, C4.17(C42⋊C2), C2.11(Q8○M4(2)), C22⋊C8.138C22, C22.170(C23×C4), (C2×C42).755C22, (C22×C8).430C22, C23.101(C22×C4), (C22×C4).913C23, C42.7C22⋊20C2, C42.6C22⋊28C2, C22.5(C42⋊C2), C42⋊C2.291C22, (C2×M4(2)).344C22, (C2×C8⋊C4)⋊32C2, C4⋊C4.218(C2×C4), (C4×C4○D4).12C2, C4.293(C2×C4○D4), (C2×D4).228(C2×C4), C22⋊C4.69(C2×C4), (C2×Q8).206(C2×C4), (C2×C4).680(C4○D4), (C2×C4).258(C22×C4), (C22×C4).336(C2×C4), C2.42(C2×C42⋊C2), (C22×C8)⋊C2.18C2, (C2×C4○D4).281C22, SmallGroup(128,1655)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.261C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >
Subgroups: 268 in 192 conjugacy classes, 132 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22×C8, C2×M4(2), C2×C4○D4, C2×C8⋊C4, C4×M4(2), (C22×C8)⋊C2, C42.6C22, C42.6C4, C42.7C22, C4×C4○D4, C42.261C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, C2×C42⋊C2, Q8○M4(2), C42.261C23
►On 64 points
(1 33 55 47)(2 44 56 38)(3 35 49 41)(4 46 50 40)(5 37 51 43)(6 48 52 34)(7 39 53 45)(8 42 54 36)(9 26 64 22)(10 19 57 31)(11 28 58 24)(12 21 59 25)(13 30 60 18)(14 23 61 27)(15 32 62 20)(16 17 63 29)
(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 17)(2 30)(3 19)(4 32)(5 21)(6 26)(7 23)(8 28)(9 34)(10 41)(11 36)(12 43)(13 38)(14 45)(15 40)(16 47)(18 56)(20 50)(22 52)(24 54)(25 51)(27 53)(29 55)(31 49)(33 63)(35 57)(37 59)(39 61)(42 58)(44 60)(46 62)(48 64)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
G:=sub<Sym(64)| (1,33,55,47)(2,44,56,38)(3,35,49,41)(4,46,50,40)(5,37,51,43)(6,48,52,34)(7,39,53,45)(8,42,54,36)(9,26,64,22)(10,19,57,31)(11,28,58,24)(12,21,59,25)(13,30,60,18)(14,23,61,27)(15,32,62,20)(16,17,63,29), (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), (1,17)(2,30)(3,19)(4,32)(5,21)(6,26)(7,23)(8,28)(9,34)(10,41)(11,36)(12,43)(13,38)(14,45)(15,40)(16,47)(18,56)(20,50)(22,52)(24,54)(25,51)(27,53)(29,55)(31,49)(33,63)(35,57)(37,59)(39,61)(42,58)(44,60)(46,62)(48,64), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)>;
G:=Group( (1,33,55,47)(2,44,56,38)(3,35,49,41)(4,46,50,40)(5,37,51,43)(6,48,52,34)(7,39,53,45)(8,42,54,36)(9,26,64,22)(10,19,57,31)(11,28,58,24)(12,21,59,25)(13,30,60,18)(14,23,61,27)(15,32,62,20)(16,17,63,29), (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), (1,17)(2,30)(3,19)(4,32)(5,21)(6,26)(7,23)(8,28)(9,34)(10,41)(11,36)(12,43)(13,38)(14,45)(15,40)(16,47)(18,56)(20,50)(22,52)(24,54)(25,51)(27,53)(29,55)(31,49)(33,63)(35,57)(37,59)(39,61)(42,58)(44,60)(46,62)(48,64), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60) );
G=PermutationGroup([[(1,33,55,47),(2,44,56,38),(3,35,49,41),(4,46,50,40),(5,37,51,43),(6,48,52,34),(7,39,53,45),(8,42,54,36),(9,26,64,22),(10,19,57,31),(11,28,58,24),(12,21,59,25),(13,30,60,18),(14,23,61,27),(15,32,62,20),(16,17,63,29)], [(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)], [(1,17),(2,30),(3,19),(4,32),(5,21),(6,26),(7,23),(8,28),(9,34),(10,41),(11,36),(12,43),(13,38),(14,45),(15,40),(16,47),(18,56),(20,50),(22,52),(24,54),(25,51),(27,53),(29,55),(31,49),(33,63),(35,57),(37,59),(39,61),(42,58),(44,60),(46,62),(48,64)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4○D4 | Q8○M4(2) |
| kernel | C42.261C23 | C2×C8⋊C4 | C4×M4(2) | (C22×C8)⋊C2 | C42.6C22 | C42.6C4 | C42.7C22 | C4×C4○D4 | C4×D4 | C4×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 12 | 4 | 8 | 4 |
Matrix representation of C42.261C23 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 6 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 14 | 13 | 0 | 0 | 0 | 0 |
| 11 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 7 |
| 0 | 0 | 10 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 | 0 | 7 |
| 0 | 0 | 7 | 0 | 7 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [13,6,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[14,11,0,0,0,0,13,3,0,0,0,0,0,0,0,10,0,7,0,0,10,0,7,0,0,0,0,7,0,7,0,0,7,0,7,0],[16,10,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C42.261C23 in GAP, Magma, Sage, TeX
C_4^2._{261}C_2^3 % in TeX
G:=Group("C4^2.261C2^3"); // GroupNames label
G:=SmallGroup(128,1655);
// by ID
G=gap.SmallGroup(128,1655);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,100,1018,124]);
// 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,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations