p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.379D4, C23.16C42, C22⋊C8⋊11C4, C4.157(C4×D4), C24.49(C2×C4), (C2×C4).23C42, C22.21(C8○D4), C22.50(C2×C42), (C23×C4).224C22, (C22×C8).470C22, (C2×C42).989C22, C23.252(C22×C4), C2.12(C8○2M4(2)), C22.7C42⋊36C2, (C22×C4).1605C23, C22.50(C42⋊C2), C2.2(C42.7C22), (C2×C4×C8)⋊6C2, (C2×C4⋊C4).46C4, (C2×C8⋊C4)⋊18C2, C2.9(C4×C22⋊C4), (C2×C8).130(C2×C4), (C2×C4).1495(C2×D4), (C2×C22⋊C4).22C4, (C2×C22⋊C8).41C2, (C2×C4).915(C4○D4), (C22×C4).106(C2×C4), (C2×C4).595(C22×C4), (C2×C4).188(C22⋊C4), (C2×C42⋊C2).12C2, C2.2((C22×C8)⋊C2), C22.113(C2×C22⋊C4), SmallGroup(128,482)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.379D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 268 in 168 conjugacy classes, 84 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C4×C8, C8⋊C4, C22⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C23×C4, C22.7C42, C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8, C2×C42⋊C2, C42.379D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C8○D4, C4×C22⋊C4, C8○2M4(2), (C22×C8)⋊C2, C42.7C22, C42.379D4
►On 64 points
(1 9 50 19)(2 14 51 24)(3 11 52 21)(4 16 53 18)(5 13 54 23)(6 10 55 20)(7 15 56 17)(8 12 49 22)(25 60 41 33)(26 57 42 38)(27 62 43 35)(28 59 44 40)(29 64 45 37)(30 61 46 34)(31 58 47 39)(32 63 48 36)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 27 13 31)(10 28 14 32)(11 29 15 25)(12 30 16 26)(17 41 21 45)(18 42 22 46)(19 43 23 47)(20 44 24 48)(49 61 53 57)(50 62 54 58)(51 63 55 59)(52 64 56 60)
(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 47 50 31)(2 16 51 18)(3 41 52 25)(4 10 53 20)(5 43 54 27)(6 12 55 22)(7 45 56 29)(8 14 49 24)(9 35 19 62)(11 37 21 64)(13 39 23 58)(15 33 17 60)(26 63 42 36)(28 57 44 38)(30 59 46 40)(32 61 48 34)
G:=sub<Sym(64)| (1,9,50,19)(2,14,51,24)(3,11,52,21)(4,16,53,18)(5,13,54,23)(6,10,55,20)(7,15,56,17)(8,12,49,22)(25,60,41,33)(26,57,42,38)(27,62,43,35)(28,59,44,40)(29,64,45,37)(30,61,46,34)(31,58,47,39)(32,63,48,36), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,27,13,31)(10,28,14,32)(11,29,15,25)(12,30,16,26)(17,41,21,45)(18,42,22,46)(19,43,23,47)(20,44,24,48)(49,61,53,57)(50,62,54,58)(51,63,55,59)(52,64,56,60), (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), (1,47,50,31)(2,16,51,18)(3,41,52,25)(4,10,53,20)(5,43,54,27)(6,12,55,22)(7,45,56,29)(8,14,49,24)(9,35,19,62)(11,37,21,64)(13,39,23,58)(15,33,17,60)(26,63,42,36)(28,57,44,38)(30,59,46,40)(32,61,48,34)>;
G:=Group( (1,9,50,19)(2,14,51,24)(3,11,52,21)(4,16,53,18)(5,13,54,23)(6,10,55,20)(7,15,56,17)(8,12,49,22)(25,60,41,33)(26,57,42,38)(27,62,43,35)(28,59,44,40)(29,64,45,37)(30,61,46,34)(31,58,47,39)(32,63,48,36), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,27,13,31)(10,28,14,32)(11,29,15,25)(12,30,16,26)(17,41,21,45)(18,42,22,46)(19,43,23,47)(20,44,24,48)(49,61,53,57)(50,62,54,58)(51,63,55,59)(52,64,56,60), (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), (1,47,50,31)(2,16,51,18)(3,41,52,25)(4,10,53,20)(5,43,54,27)(6,12,55,22)(7,45,56,29)(8,14,49,24)(9,35,19,62)(11,37,21,64)(13,39,23,58)(15,33,17,60)(26,63,42,36)(28,57,44,38)(30,59,46,40)(32,61,48,34) );
G=PermutationGroup([[(1,9,50,19),(2,14,51,24),(3,11,52,21),(4,16,53,18),(5,13,54,23),(6,10,55,20),(7,15,56,17),(8,12,49,22),(25,60,41,33),(26,57,42,38),(27,62,43,35),(28,59,44,40),(29,64,45,37),(30,61,46,34),(31,58,47,39),(32,63,48,36)], [(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,27,13,31),(10,28,14,32),(11,29,15,25),(12,30,16,26),(17,41,21,45),(18,42,22,46),(19,43,23,47),(20,44,24,48),(49,61,53,57),(50,62,54,58),(51,63,55,59),(52,64,56,60)], [(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)], [(1,47,50,31),(2,16,51,18),(3,41,52,25),(4,10,53,20),(5,43,54,27),(6,12,55,22),(7,45,56,29),(8,14,49,24),(9,35,19,62),(11,37,21,64),(13,39,23,58),(15,33,17,60),(26,63,42,36),(28,57,44,38),(30,59,46,40),(32,61,48,34)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4V | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 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 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | C8○D4 |
| kernel | C42.379D4 | C22.7C42 | C2×C4×C8 | C2×C8⋊C4 | C2×C22⋊C8 | C2×C42⋊C2 | C22⋊C8 | C2×C22⋊C4 | C2×C4⋊C4 | C42 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 4 | 4 | 4 | 4 | 16 |
Matrix representation of C42.379D4 ►in GL5(𝔽17)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 2 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 15 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 4 | 0 | 0 |
| 0 | 9 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(17))| [4,0,0,0,0,0,16,15,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13],[13,0,0,0,0,0,2,0,0,0,0,15,15,0,0,0,0,0,0,15,0,0,0,2,0],[4,0,0,0,0,0,13,9,0,0,0,4,4,0,0,0,0,0,0,4,0,0,0,13,0] >;
C42.379D4 in GAP, Magma, Sage, TeX
C_4^2._{379}D_4 % in TeX
G:=Group("C4^2.379D4"); // GroupNames label
G:=SmallGroup(128,482);
// by ID
G=gap.SmallGroup(128,482);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,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=a^2*b^-1*c^3>;
// generators/relations