p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.378D4, C23.34C42, C23.29M4(2), C22⋊C8⋊10C4, C4.156(C4×D4), (C2×C4).44C42, (C23×C4).28C4, (C2×C42).33C4, C22⋊1(C8⋊C4), C24.110(C2×C4), C2.10(C4×M4(2)), (C2×C4).59M4(2), (C22×C42).6C2, C22.49(C2×C42), C2.2(C24.4C4), C2.2(C42.6C4), (C22×C8).377C22, (C2×C42).988C22, C23.251(C22×C4), (C23×C4).629C22, C22.37(C2×M4(2)), C22.7C42⋊35C2, (C22×C4).1604C23, C22.49(C42⋊C2), (C2×C8)⋊23(C2×C4), C2.7(C2×C8⋊C4), (C2×C8⋊C4)⋊17C2, C2.8(C4×C22⋊C4), (C2×C4).1494(C2×D4), (C2×C22⋊C8).40C2, (C2×C4).914(C4○D4), (C2×C4).594(C22×C4), (C22×C4).434(C2×C4), (C2×C4).325(C22⋊C4), C22.112(C2×C22⋊C4), SmallGroup(128,481)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.378D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b-1, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 316 in 206 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C8⋊C4, C22⋊C8, C2×C42, C2×C42, C2×C42, C22×C8, C23×C4, C23×C4, C22.7C42, C2×C8⋊C4, C2×C22⋊C8, C22×C42, C42.378D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, M4(2), C22×C4, C2×D4, C4○D4, C8⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C2×M4(2), C4×C22⋊C4, C2×C8⋊C4, C4×M4(2), C24.4C4, C42.6C4, C42.378D4
►On 64 points
(1 57 51 14)(2 62 52 11)(3 59 53 16)(4 64 54 13)(5 61 55 10)(6 58 56 15)(7 63 49 12)(8 60 50 9)(17 35 25 45)(18 40 26 42)(19 37 27 47)(20 34 28 44)(21 39 29 41)(22 36 30 46)(23 33 31 43)(24 38 32 48)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 44 13 48)(10 45 14 41)(11 46 15 42)(12 47 16 43)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 63 37 59)(34 64 38 60)(35 57 39 61)(36 58 40 62)
(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 11 5 40 29 15)(2 57 26 45 6 61 30 41)(3 38 27 13 7 34 31 9)(4 59 28 47 8 63 32 43)(10 22 39 52 14 18 35 56)(12 24 33 54 16 20 37 50)(17 62 55 42 21 58 51 46)(19 64 49 44 23 60 53 48)
G:=sub<Sym(64)| (1,57,51,14)(2,62,52,11)(3,59,53,16)(4,64,54,13)(5,61,55,10)(6,58,56,15)(7,63,49,12)(8,60,50,9)(17,35,25,45)(18,40,26,42)(19,37,27,47)(20,34,28,44)(21,39,29,41)(22,36,30,46)(23,33,31,43)(24,38,32,48), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,44,13,48)(10,45,14,41)(11,46,15,42)(12,47,16,43)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,63,37,59)(34,64,38,60)(35,57,39,61)(36,58,40,62), (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), (1,36,25,11,5,40,29,15)(2,57,26,45,6,61,30,41)(3,38,27,13,7,34,31,9)(4,59,28,47,8,63,32,43)(10,22,39,52,14,18,35,56)(12,24,33,54,16,20,37,50)(17,62,55,42,21,58,51,46)(19,64,49,44,23,60,53,48)>;
G:=Group( (1,57,51,14)(2,62,52,11)(3,59,53,16)(4,64,54,13)(5,61,55,10)(6,58,56,15)(7,63,49,12)(8,60,50,9)(17,35,25,45)(18,40,26,42)(19,37,27,47)(20,34,28,44)(21,39,29,41)(22,36,30,46)(23,33,31,43)(24,38,32,48), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,44,13,48)(10,45,14,41)(11,46,15,42)(12,47,16,43)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,63,37,59)(34,64,38,60)(35,57,39,61)(36,58,40,62), (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), (1,36,25,11,5,40,29,15)(2,57,26,45,6,61,30,41)(3,38,27,13,7,34,31,9)(4,59,28,47,8,63,32,43)(10,22,39,52,14,18,35,56)(12,24,33,54,16,20,37,50)(17,62,55,42,21,58,51,46)(19,64,49,44,23,60,53,48) );
G=PermutationGroup([[(1,57,51,14),(2,62,52,11),(3,59,53,16),(4,64,54,13),(5,61,55,10),(6,58,56,15),(7,63,49,12),(8,60,50,9),(17,35,25,45),(18,40,26,42),(19,37,27,47),(20,34,28,44),(21,39,29,41),(22,36,30,46),(23,33,31,43),(24,38,32,48)], [(1,21,5,17),(2,22,6,18),(3,23,7,19),(4,24,8,20),(9,44,13,48),(10,45,14,41),(11,46,15,42),(12,47,16,43),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,63,37,59),(34,64,38,60),(35,57,39,61),(36,58,40,62)], [(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)], [(1,36,25,11,5,40,29,15),(2,57,26,45,6,61,30,41),(3,38,27,13,7,34,31,9),(4,59,28,47,8,63,32,43),(10,22,39,52,14,18,35,56),(12,24,33,54,16,20,37,50),(17,62,55,42,21,58,51,46),(19,64,49,44,23,60,53,48)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | C4○D4 | M4(2) |
| kernel | C42.378D4 | C22.7C42 | C2×C8⋊C4 | C2×C22⋊C8 | C22×C42 | C22⋊C8 | C2×C42 | C23×C4 | C42 | C2×C4 | C2×C4 | C23 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 4 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C42.378D4 ►in GL5(𝔽17)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 16 |
| 0 | 0 | 0 | 5 | 2 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 13 |
| 0 | 0 | 0 | 12 | 8 |
G:=sub<GL(5,GF(17))| [4,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,15,5,0,0,0,16,2],[13,0,0,0,0,0,0,13,0,0,0,16,0,0,0,0,0,0,9,12,0,0,0,13,8] >;
C42.378D4 in GAP, Magma, Sage, TeX
C_4^2._{378}D_4 % in TeX
G:=Group("C4^2.378D4"); // GroupNames label
G:=SmallGroup(128,481);
// by ID
G=gap.SmallGroup(128,481);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations