p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.433D4, C4.11(C4×D4), (C2×C8).230D4, C42⋊8C4⋊5C2, C4.4D4⋊14C4, C4.74(C4⋊D4), C42.267(C2×C4), C23.797(C2×D4), (C22×C4).557D4, C2.2(C8.12D4), C22.74(C4○D8), C22.33(C4⋊1D4), (C22×C8).491C22, (C22×D4).42C22, (C22×Q8).33C22, (C2×C42).1073C22, (C22×C4).1402C23, C22.64(C4.4D4), C2.28(C23.24D4), C2.10(C24.3C22), C2.4(C42.78C22), (C2×C4×C8)⋊14C2, (C2×Q8⋊C4)⋊8C2, (C2×C4).736(C2×D4), (C2×Q8).89(C2×C4), (C2×D4⋊C4).8C2, (C2×D4).104(C2×C4), (C2×C4⋊C4).85C22, (C2×C4.4D4).7C2, (C2×C4).593(C4○D4), (C2×C4).416(C22×C4), (C2×C4).199(C22⋊C4), C22.280(C2×C22⋊C4), SmallGroup(128,690)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.433D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 372 in 168 conjugacy classes, 64 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C4×C8, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C4.4D4, C22×C8, C22×D4, C22×Q8, C42⋊8C4, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.4D4, C42.433D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4○D8, C24.3C22, C23.24D4, C42.78C22, C8.12D4, C42.433D4
►On 64 points
(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 36 25 39)(2 33 26 40)(3 34 27 37)(4 35 28 38)(5 41 61 45)(6 42 62 46)(7 43 63 47)(8 44 64 48)(9 29 13 50)(10 30 14 51)(11 31 15 52)(12 32 16 49)(17 56 24 59)(18 53 21 60)(19 54 22 57)(20 55 23 58)
(1 59 50 8)(2 55 51 63)(3 57 52 6)(4 53 49 61)(5 28 60 32)(7 26 58 30)(9 48 36 24)(10 43 33 20)(11 46 34 22)(12 41 35 18)(13 44 39 17)(14 47 40 23)(15 42 37 19)(16 45 38 21)(25 56 29 64)(27 54 31 62)
(1 4 25 28)(2 27 26 3)(5 24 61 17)(6 20 62 23)(7 22 63 19)(8 18 64 21)(9 16 13 12)(10 11 14 15)(29 32 50 49)(30 52 51 31)(33 34 40 37)(35 36 38 39)(41 56 45 59)(42 58 46 55)(43 54 47 57)(44 60 48 53)
G:=sub<Sym(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), (1,36,25,39)(2,33,26,40)(3,34,27,37)(4,35,28,38)(5,41,61,45)(6,42,62,46)(7,43,63,47)(8,44,64,48)(9,29,13,50)(10,30,14,51)(11,31,15,52)(12,32,16,49)(17,56,24,59)(18,53,21,60)(19,54,22,57)(20,55,23,58), (1,59,50,8)(2,55,51,63)(3,57,52,6)(4,53,49,61)(5,28,60,32)(7,26,58,30)(9,48,36,24)(10,43,33,20)(11,46,34,22)(12,41,35,18)(13,44,39,17)(14,47,40,23)(15,42,37,19)(16,45,38,21)(25,56,29,64)(27,54,31,62), (1,4,25,28)(2,27,26,3)(5,24,61,17)(6,20,62,23)(7,22,63,19)(8,18,64,21)(9,16,13,12)(10,11,14,15)(29,32,50,49)(30,52,51,31)(33,34,40,37)(35,36,38,39)(41,56,45,59)(42,58,46,55)(43,54,47,57)(44,60,48,53)>;
G:=Group( (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), (1,36,25,39)(2,33,26,40)(3,34,27,37)(4,35,28,38)(5,41,61,45)(6,42,62,46)(7,43,63,47)(8,44,64,48)(9,29,13,50)(10,30,14,51)(11,31,15,52)(12,32,16,49)(17,56,24,59)(18,53,21,60)(19,54,22,57)(20,55,23,58), (1,59,50,8)(2,55,51,63)(3,57,52,6)(4,53,49,61)(5,28,60,32)(7,26,58,30)(9,48,36,24)(10,43,33,20)(11,46,34,22)(12,41,35,18)(13,44,39,17)(14,47,40,23)(15,42,37,19)(16,45,38,21)(25,56,29,64)(27,54,31,62), (1,4,25,28)(2,27,26,3)(5,24,61,17)(6,20,62,23)(7,22,63,19)(8,18,64,21)(9,16,13,12)(10,11,14,15)(29,32,50,49)(30,52,51,31)(33,34,40,37)(35,36,38,39)(41,56,45,59)(42,58,46,55)(43,54,47,57)(44,60,48,53) );
G=PermutationGroup([[(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)], [(1,36,25,39),(2,33,26,40),(3,34,27,37),(4,35,28,38),(5,41,61,45),(6,42,62,46),(7,43,63,47),(8,44,64,48),(9,29,13,50),(10,30,14,51),(11,31,15,52),(12,32,16,49),(17,56,24,59),(18,53,21,60),(19,54,22,57),(20,55,23,58)], [(1,59,50,8),(2,55,51,63),(3,57,52,6),(4,53,49,61),(5,28,60,32),(7,26,58,30),(9,48,36,24),(10,43,33,20),(11,46,34,22),(12,41,35,18),(13,44,39,17),(14,47,40,23),(15,42,37,19),(16,45,38,21),(25,56,29,64),(27,54,31,62)], [(1,4,25,28),(2,27,26,3),(5,24,61,17),(6,20,62,23),(7,22,63,19),(8,18,64,21),(9,16,13,12),(10,11,14,15),(29,32,50,49),(30,52,51,31),(33,34,40,37),(35,36,38,39),(41,56,45,59),(42,58,46,55),(43,54,47,57),(44,60,48,53)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C4○D8 |
| kernel | C42.433D4 | C42⋊8C4 | C2×C4×C8 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.4D4 | C4.4D4 | C42 | C2×C8 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 2 | 4 | 2 | 4 | 16 |
Matrix representation of C42.433D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 3 | 14 | 0 | 0 |
| 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 13 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,3,14,0,0,0,14,14,0,0,0,0,0,0,13,0,0,0,13,0],[16,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,1,0] >;
C42.433D4 in GAP, Magma, Sage, TeX
C_4^2._{433}D_4 % in TeX
G:=Group("C4^2.433D4"); // GroupNames label
G:=SmallGroup(128,690);
// by ID
G=gap.SmallGroup(128,690);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;
// generators/relations