p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.291C23, (C8×D4)⋊41C2, (C8×Q8)⋊30C2, C8○3(C4⋊D4), C8○2(C8⋊9D4), C8○2(C8⋊6D4), C8⋊9D4⋊54C2, C8⋊6D4⋊52C2, C8○2(C8⋊4Q8), C8⋊4Q8⋊52C2, C8○3(C22⋊Q8), C8○(C42⋊2C2), C4.20(C8○D4), C8○2(C4.4D4), C4⋊D4.36C4, C8○2(C42.C2), C22⋊Q8.36C4, C8.106(C4○D4), C4⋊C8.362C22, C8○(C42.6C4), (C2×C4).663C24, C42⋊2C2.8C4, C42.287(C2×C4), (C4×C8).438C22, (C2×C8).641C23, C4.4D4.28C4, C22.3(C8○D4), C42.C2.28C4, C8○2M4(2)⋊33C2, C42.6C4⋊59C2, (C4×D4).294C22, C8○2(C22.D4), C23.38(C22×C4), (C4×Q8).279C22, C8⋊C4.163C22, C22⋊C8.233C22, C8○(C42.7C22), (C22×C8).516C22, C22.188(C23×C4), C22.D4.16C4, C8○(C23.36C23), (C2×C42).1120C22, C42.7C22⋊34C2, (C22×C4).1278C23, C42⋊C2.306C22, (C2×M4(2)).365C22, C23.36C23.35C2, (C2×C4×C8)⋊45C2, (C2×C8)○(C8⋊4Q8), C2.24(C2×C8○D4), C2.45(C4×C4○D4), C4⋊C4.164(C2×C4), (C2×C8)○(C4.4D4), C4.314(C2×C4○D4), (C2×C8)○(C42.C2), (C2×C8)○(C42⋊2C2), (C2×D4).180(C2×C4), C22⋊C4.41(C2×C4), (C2×C4).77(C22×C4), (C2×Q8).164(C2×C4), (C2×C8)○(C42.6C4), (C22×C4).388(C2×C4), (C2×C8)○(C22.D4), (C2×C8)○(C42.7C22), (C2×C8)○(C23.36C23), SmallGroup(128,1698)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.291C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b-1, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ce=ec, ede-1=b2d >
Subgroups: 252 in 190 conjugacy classes, 132 normal (52 characteristic)
C1, C2, C2, C4, 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, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C22×C8, C2×M4(2), C2×C4×C8, C8○2M4(2), C42.6C4, C42.7C22, C8×D4, C8×D4, C8⋊9D4, C8⋊6D4, C8×Q8, C8⋊4Q8, C23.36C23, C42.291C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C8○D4, C23×C4, C2×C4○D4, C4×C4○D4, C2×C8○D4, C42.291C23
►On 64 points
(1 33 20 48)(2 34 21 41)(3 35 22 42)(4 36 23 43)(5 37 24 44)(6 38 17 45)(7 39 18 46)(8 40 19 47)(9 51 28 60)(10 52 29 61)(11 53 30 62)(12 54 31 63)(13 55 32 64)(14 56 25 57)(15 49 26 58)(16 50 27 59)
(1 18 5 22)(2 19 6 23)(3 20 7 24)(4 21 8 17)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)
(1 48 5 44)(2 38 6 34)(3 42 7 46)(4 40 8 36)(9 64 13 60)(10 52 14 56)(11 58 15 62)(12 54 16 50)(17 41 21 45)(18 39 22 35)(19 43 23 47)(20 33 24 37)(25 57 29 61)(26 53 30 49)(27 59 31 63)(28 55 32 51)
(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 59 24 54)(2 64 17 51)(3 61 18 56)(4 58 19 53)(5 63 20 50)(6 60 21 55)(7 57 22 52)(8 62 23 49)(9 45 32 34)(10 42 25 39)(11 47 26 36)(12 44 27 33)(13 41 28 38)(14 46 29 35)(15 43 30 40)(16 48 31 37)
G:=sub<Sym(64)| (1,33,20,48)(2,34,21,41)(3,35,22,42)(4,36,23,43)(5,37,24,44)(6,38,17,45)(7,39,18,46)(8,40,19,47)(9,51,28,60)(10,52,29,61)(11,53,30,62)(12,54,31,63)(13,55,32,64)(14,56,25,57)(15,49,26,58)(16,50,27,59), (1,18,5,22)(2,19,6,23)(3,20,7,24)(4,21,8,17)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63), (1,48,5,44)(2,38,6,34)(3,42,7,46)(4,40,8,36)(9,64,13,60)(10,52,14,56)(11,58,15,62)(12,54,16,50)(17,41,21,45)(18,39,22,35)(19,43,23,47)(20,33,24,37)(25,57,29,61)(26,53,30,49)(27,59,31,63)(28,55,32,51), (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), (1,59,24,54)(2,64,17,51)(3,61,18,56)(4,58,19,53)(5,63,20,50)(6,60,21,55)(7,57,22,52)(8,62,23,49)(9,45,32,34)(10,42,25,39)(11,47,26,36)(12,44,27,33)(13,41,28,38)(14,46,29,35)(15,43,30,40)(16,48,31,37)>;
G:=Group( (1,33,20,48)(2,34,21,41)(3,35,22,42)(4,36,23,43)(5,37,24,44)(6,38,17,45)(7,39,18,46)(8,40,19,47)(9,51,28,60)(10,52,29,61)(11,53,30,62)(12,54,31,63)(13,55,32,64)(14,56,25,57)(15,49,26,58)(16,50,27,59), (1,18,5,22)(2,19,6,23)(3,20,7,24)(4,21,8,17)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63), (1,48,5,44)(2,38,6,34)(3,42,7,46)(4,40,8,36)(9,64,13,60)(10,52,14,56)(11,58,15,62)(12,54,16,50)(17,41,21,45)(18,39,22,35)(19,43,23,47)(20,33,24,37)(25,57,29,61)(26,53,30,49)(27,59,31,63)(28,55,32,51), (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), (1,59,24,54)(2,64,17,51)(3,61,18,56)(4,58,19,53)(5,63,20,50)(6,60,21,55)(7,57,22,52)(8,62,23,49)(9,45,32,34)(10,42,25,39)(11,47,26,36)(12,44,27,33)(13,41,28,38)(14,46,29,35)(15,43,30,40)(16,48,31,37) );
G=PermutationGroup([[(1,33,20,48),(2,34,21,41),(3,35,22,42),(4,36,23,43),(5,37,24,44),(6,38,17,45),(7,39,18,46),(8,40,19,47),(9,51,28,60),(10,52,29,61),(11,53,30,62),(12,54,31,63),(13,55,32,64),(14,56,25,57),(15,49,26,58),(16,50,27,59)], [(1,18,5,22),(2,19,6,23),(3,20,7,24),(4,21,8,17),(9,26,13,30),(10,27,14,31),(11,28,15,32),(12,29,16,25),(33,46,37,42),(34,47,38,43),(35,48,39,44),(36,41,40,45),(49,64,53,60),(50,57,54,61),(51,58,55,62),(52,59,56,63)], [(1,48,5,44),(2,38,6,34),(3,42,7,46),(4,40,8,36),(9,64,13,60),(10,52,14,56),(11,58,15,62),(12,54,16,50),(17,41,21,45),(18,39,22,35),(19,43,23,47),(20,33,24,37),(25,57,29,61),(26,53,30,49),(27,59,31,63),(28,55,32,51)], [(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)], [(1,59,24,54),(2,64,17,51),(3,61,18,56),(4,58,19,53),(5,63,20,50),(6,60,21,55),(7,57,22,52),(8,62,23,49),(9,45,32,34),(10,42,25,39),(11,47,26,36),(12,44,27,33),(13,41,28,38),(14,46,29,35),(15,43,30,40),(16,48,31,37)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8H | 8I | ··· | 8T | 8U | ··· | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4○D4 | C8○D4 | C8○D4 |
| kernel | C42.291C23 | C2×C4×C8 | C8○2M4(2) | C42.6C4 | C42.7C22 | C8×D4 | C8⋊9D4 | C8⋊6D4 | C8×Q8 | C8⋊4Q8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C8 | C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 3 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 8 |
Matrix representation of C42.291C23 ►in GL4(𝔽17) generated by
| 1 | 16 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 8 | 0 |
| 16 | 1 | 0 | 0 |
| 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [1,2,0,0,16,16,0,0,0,0,0,8,0,0,15,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[16,15,0,0,0,1,0,0,0,0,0,8,0,0,2,0],[16,15,0,0,1,1,0,0,0,0,0,16,0,0,13,0],[4,0,0,0,0,4,0,0,0,0,0,9,0,0,15,0] >;
C42.291C23 in GAP, Magma, Sage, TeX
C_4^2._{291}C_2^3 % in TeX
G:=Group("C4^2.291C2^3"); // GroupNames label
G:=SmallGroup(128,1698);
// by ID
G=gap.SmallGroup(128,1698);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2*b^-1,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*b^2*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations