p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.42Q8, C42.456D4, C42.621C23, (C2×C8)⋊6C8, C8.17(C2×C8), (C4×C8).26C4, C4○2(C8⋊2C8), C4○2(C8⋊1C8), C8⋊1C8⋊31C2, C8⋊2C8⋊31C2, C4.27(C4⋊C8), C42○(C8⋊1C8), C42○(C8⋊2C8), C22.5(C4⋊C8), (C22×C8).36C4, C4.26(C22×C8), C4.122(C4○D8), (C22×C4).74Q8, C4⋊C8.265C22, C23.47(C4⋊C4), C42.309(C2×C4), (C4×C8).389C22, (C22×C4).541D4, C4.41(C2×M4(2)), (C2×C4).75M4(2), C4.14(C8.C4), C42.12C4.28C2, (C2×C42).1040C22, C2.1(C23.25D4), C2.6(C2×C4⋊C8), (C2×C4×C8).41C2, (C2×C4).82(C2×C8), (C2×C8).219(C2×C4), C2.3(C2×C8.C4), C22.47(C2×C4⋊C4), (C2×C4).148(C2×Q8), (C2×C4).161(C4⋊C4), (C2×C4).1457(C2×D4), (C22×C4).473(C2×C4), (C2×C4).503(C22×C4), SmallGroup(128,296)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.42Q8
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 124 in 90 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C2×C8, C22×C4, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C22×C8, C8⋊2C8, C8⋊1C8, C2×C4×C8, C42.12C4, C42.42Q8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C8.C4, C2×C4⋊C4, C22×C8, C2×M4(2), C4○D8, C2×C4⋊C8, C23.25D4, C2×C8.C4, C42.42Q8
►On 64 points
(1 7 5 3)(2 8 6 4)(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 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)
(1 33 41 55)(2 34 42 56)(3 35 43 49)(4 36 44 50)(5 37 45 51)(6 38 46 52)(7 39 47 53)(8 40 48 54)(9 57 25 23)(10 58 26 24)(11 59 27 17)(12 60 28 18)(13 61 29 19)(14 62 30 20)(15 63 31 21)(16 64 32 22)
(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 32 39 20 45 12 49 58)(2 27 40 23 46 15 50 61)(3 30 33 18 47 10 51 64)(4 25 34 21 48 13 52 59)(5 28 35 24 41 16 53 62)(6 31 36 19 42 11 54 57)(7 26 37 22 43 14 55 60)(8 29 38 17 44 9 56 63)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64), (1,33,41,55)(2,34,42,56)(3,35,43,49)(4,36,44,50)(5,37,45,51)(6,38,46,52)(7,39,47,53)(8,40,48,54)(9,57,25,23)(10,58,26,24)(11,59,27,17)(12,60,28,18)(13,61,29,19)(14,62,30,20)(15,63,31,21)(16,64,32,22), (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), (1,32,39,20,45,12,49,58)(2,27,40,23,46,15,50,61)(3,30,33,18,47,10,51,64)(4,25,34,21,48,13,52,59)(5,28,35,24,41,16,53,62)(6,31,36,19,42,11,54,57)(7,26,37,22,43,14,55,60)(8,29,38,17,44,9,56,63)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64), (1,33,41,55)(2,34,42,56)(3,35,43,49)(4,36,44,50)(5,37,45,51)(6,38,46,52)(7,39,47,53)(8,40,48,54)(9,57,25,23)(10,58,26,24)(11,59,27,17)(12,60,28,18)(13,61,29,19)(14,62,30,20)(15,63,31,21)(16,64,32,22), (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), (1,32,39,20,45,12,49,58)(2,27,40,23,46,15,50,61)(3,30,33,18,47,10,51,64)(4,25,34,21,48,13,52,59)(5,28,35,24,41,16,53,62)(6,31,36,19,42,11,54,57)(7,26,37,22,43,14,55,60)(8,29,38,17,44,9,56,63) );
G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,59,61,63),(58,60,62,64)], [(1,33,41,55),(2,34,42,56),(3,35,43,49),(4,36,44,50),(5,37,45,51),(6,38,46,52),(7,39,47,53),(8,40,48,54),(9,57,25,23),(10,58,26,24),(11,59,27,17),(12,60,28,18),(13,61,29,19),(14,62,30,20),(15,63,31,21),(16,64,32,22)], [(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)], [(1,32,39,20,45,12,49,58),(2,27,40,23,46,15,50,61),(3,30,33,18,47,10,51,64),(4,25,34,21,48,13,52,59),(5,28,35,24,41,16,53,62),(6,31,36,19,42,11,54,57),(7,26,37,22,43,14,55,60),(8,29,38,17,44,9,56,63)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D4 | Q8 | M4(2) | C8.C4 | C4○D8 |
| kernel | C42.42Q8 | C8⋊2C8 | C8⋊1C8 | C2×C4×C8 | C42.12C4 | C4×C8 | C22×C8 | C2×C8 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 4 | 8 | 8 |
Matrix representation of C42.42Q8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 13 |
| 1 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 9 |
| 15 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(17))| [16,0,0,0,13,0,0,0,13],[4,0,0,0,4,0,0,0,13],[1,0,0,0,15,0,0,0,9],[15,0,0,0,0,1,0,1,0] >;
C42.42Q8 in GAP, Magma, Sage, TeX
C_4^2._{42}Q_8 % in TeX
G:=Group("C4^2.42Q8"); // GroupNames label
G:=SmallGroup(128,296);
// by ID
G=gap.SmallGroup(128,296);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,184,1123,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations