p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.290C23, C8⋊13(C4○D4), C4○2(C8⋊6D4), C4○2(C8⋊9D4), C8⋊6D4⋊51C2, C8⋊9D4⋊53C2, C4○2(C8⋊4Q8), C8⋊4Q8⋊51C2, (C4×D4).29C4, (C2×C4)⋊9M4(2), (C4×Q8).27C4, C4.64(C8○D4), C42○(C8⋊9D4), C42○(C8⋊6D4), C42○(C8⋊4Q8), C4⋊C8.361C22, (C4×M4(2))⋊35C2, (C4×C8).437C22, (C2×C8).427C23, (C2×C4).662C24, C42.249(C2×C4), C4.14(C2×M4(2)), C42⋊C2.31C4, (C4×D4).293C22, (C4×Q8).278C22, C8⋊C4.162C22, C42.12C4⋊49C2, C22.3(C2×M4(2)), C22⋊C8.232C22, C22.187(C23×C4), (C22×C8).585C22, (C22×C4).934C23, C23.146(C22×C4), C2.15(C22×M4(2)), (C2×C42).1119C22, (C2×M4(2)).364C22, (C2×C4×C8)⋊44C2, C2.44(C4×C4○D4), C2.23(C2×C8○D4), C4⋊C4.225(C2×C4), (C2×C4○D4).25C4, (C4×C4○D4).15C2, C4.313(C2×C4○D4), (C2×D4).232(C2×C4), C22⋊C4.74(C2×C4), (C2×Q8).209(C2×C4), (C22×C4).420(C2×C4), (C2×C4).271(C22×C4), SmallGroup(128,1697)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.290C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b-1, ab=ba, cac=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, ede=b2d >
Subgroups: 276 in 206 conjugacy classes, 140 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×M4(2), C2×C4○D4, C2×C4×C8, C4×M4(2), C42.12C4, C42.12C4, C8⋊9D4, C8⋊6D4, C8⋊4Q8, C4×C4○D4, C42.290C23
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, C24, C2×M4(2), C8○D4, C23×C4, C2×C4○D4, C4×C4○D4, C22×M4(2), C2×C8○D4, C42.290C23
►On 64 points
(1 38 25 12)(2 35 26 9)(3 40 27 14)(4 37 28 11)(5 34 29 16)(6 39 30 13)(7 36 31 10)(8 33 32 15)(17 47 55 60)(18 44 56 57)(19 41 49 62)(20 46 50 59)(21 43 51 64)(22 48 52 61)(23 45 53 58)(24 42 54 63)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 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 5)(3 7)(10 14)(12 16)(17 55)(18 52)(19 49)(20 54)(21 51)(22 56)(23 53)(24 50)(25 29)(27 31)(34 38)(36 40)(41 62)(42 59)(43 64)(44 61)(45 58)(46 63)(47 60)(48 57)
G:=sub<Sym(64)| (1,38,25,12)(2,35,26,9)(3,40,27,14)(4,37,28,11)(5,34,29,16)(6,39,30,13)(7,36,31,10)(8,33,32,15)(17,47,55,60)(18,44,56,57)(19,41,49,62)(20,46,50,59)(21,43,51,64)(22,48,52,61)(23,45,53,58)(24,42,54,63), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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), (1,5)(3,7)(10,14)(12,16)(17,55)(18,52)(19,49)(20,54)(21,51)(22,56)(23,53)(24,50)(25,29)(27,31)(34,38)(36,40)(41,62)(42,59)(43,64)(44,61)(45,58)(46,63)(47,60)(48,57)>;
G:=Group( (1,38,25,12)(2,35,26,9)(3,40,27,14)(4,37,28,11)(5,34,29,16)(6,39,30,13)(7,36,31,10)(8,33,32,15)(17,47,55,60)(18,44,56,57)(19,41,49,62)(20,46,50,59)(21,43,51,64)(22,48,52,61)(23,45,53,58)(24,42,54,63), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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), (1,5)(3,7)(10,14)(12,16)(17,55)(18,52)(19,49)(20,54)(21,51)(22,56)(23,53)(24,50)(25,29)(27,31)(34,38)(36,40)(41,62)(42,59)(43,64)(44,61)(45,58)(46,63)(47,60)(48,57) );
G=PermutationGroup([[(1,38,25,12),(2,35,26,9),(3,40,27,14),(4,37,28,11),(5,34,29,16),(6,39,30,13),(7,36,31,10),(8,33,32,15),(17,47,55,60),(18,44,56,57),(19,41,49,62),(20,46,50,59),(21,43,51,64),(22,48,52,61),(23,45,53,58),(24,42,54,63)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,41),(7,42),(8,43),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64)], [(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)], [(1,5),(3,7),(10,14),(12,16),(17,55),(18,52),(19,49),(20,54),(21,51),(22,56),(23,53),(24,50),(25,29),(27,31),(34,38),(36,40),(41,62),(42,59),(43,64),(44,61),(45,58),(46,63),(47,60),(48,57)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4R | 4S | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4○D4 | M4(2) | C8○D4 |
| kernel | C42.290C23 | C2×C4×C8 | C4×M4(2) | C42.12C4 | C8⋊9D4 | C8⋊6D4 | C8⋊4Q8 | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 3 | 4 | 2 | 2 | 1 | 6 | 6 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of C42.290C23 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 10 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 1 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,1,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,10,0,0,9,13],[1,0,0,0,0,16,0,0,0,0,16,16,0,0,0,1] >;
C42.290C23 in GAP, Magma, Sage, TeX
C_4^2._{290}C_2^3 % in TeX
G:=Group("C4^2.290C2^3"); // GroupNames label
G:=SmallGroup(128,1697);
// by ID
G=gap.SmallGroup(128,1697);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^-1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^2,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=b^2*d>;
// generators/relations