p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.168D4, C24.29C23, C23.445C24, C22.1782- 1+4, (C2×Q8).167D4, C4.59(C4⋊D4), C2.38(Q8⋊5D4), C23.Q8⋊28C2, (C2×C42).550C22, (C22×C4).537C23, C22.296(C22×D4), C24.C22⋊79C2, (C22×D4).166C22, (C22×Q8).433C22, C23.67C23⋊59C2, C24.3C22.45C2, C2.C42.183C22, C2.32(C22.26C24), C2.22(C23.38C23), C2.37(C22.50C24), (C4×C4⋊C4)⋊89C2, (C2×C4×Q8)⋊23C2, (C2×C4⋊Q8)⋊13C2, (C2×C4).354(C2×D4), C2.37(C2×C4⋊D4), (C2×C4).148(C4○D4), (C2×C4⋊C4).301C22, (C2×C4.4D4).24C2, C22.322(C2×C4○D4), (C2×C22⋊C4).179C22, SmallGroup(128,1277)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.168D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 532 in 288 conjugacy classes, 112 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4.4D4, C4⋊Q8, C22×D4, C22×Q8, C4×C4⋊C4, C24.C22, C24.3C22, C23.67C23, C23.Q8, C2×C4×Q8, C2×C4.4D4, C2×C4⋊Q8, C42.168D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4⋊D4, C22.26C24, C23.38C23, Q8⋊5D4, C22.50C24, C42.168D4
►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 15 21 11)(2 16 22 12)(3 13 23 9)(4 14 24 10)(5 56 38 26)(6 53 39 27)(7 54 40 28)(8 55 37 25)(17 30 50 60)(18 31 51 57)(19 32 52 58)(20 29 49 59)(33 47 61 44)(34 48 62 41)(35 45 63 42)(36 46 64 43)
(1 42 37 59)(2 41 38 58)(3 44 39 57)(4 43 40 60)(5 32 22 48)(6 31 23 47)(7 30 24 46)(8 29 21 45)(9 61 53 51)(10 64 54 50)(11 63 55 49)(12 62 56 52)(13 33 27 18)(14 36 28 17)(15 35 25 20)(16 34 26 19)
(1 2 21 22)(3 4 23 24)(5 37 38 8)(6 7 39 40)(9 14 13 10)(11 16 15 12)(17 33 50 61)(18 64 51 36)(19 35 52 63)(20 62 49 34)(25 56 55 26)(27 54 53 28)(29 48 59 41)(30 44 60 47)(31 46 57 43)(32 42 58 45)
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,47,61,44)(34,48,62,41)(35,45,63,42)(36,46,64,43), (1,42,37,59)(2,41,38,58)(3,44,39,57)(4,43,40,60)(5,32,22,48)(6,31,23,47)(7,30,24,46)(8,29,21,45)(9,61,53,51)(10,64,54,50)(11,63,55,49)(12,62,56,52)(13,33,27,18)(14,36,28,17)(15,35,25,20)(16,34,26,19), (1,2,21,22)(3,4,23,24)(5,37,38,8)(6,7,39,40)(9,14,13,10)(11,16,15,12)(17,33,50,61)(18,64,51,36)(19,35,52,63)(20,62,49,34)(25,56,55,26)(27,54,53,28)(29,48,59,41)(30,44,60,47)(31,46,57,43)(32,42,58,45)>;
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,47,61,44)(34,48,62,41)(35,45,63,42)(36,46,64,43), (1,42,37,59)(2,41,38,58)(3,44,39,57)(4,43,40,60)(5,32,22,48)(6,31,23,47)(7,30,24,46)(8,29,21,45)(9,61,53,51)(10,64,54,50)(11,63,55,49)(12,62,56,52)(13,33,27,18)(14,36,28,17)(15,35,25,20)(16,34,26,19), (1,2,21,22)(3,4,23,24)(5,37,38,8)(6,7,39,40)(9,14,13,10)(11,16,15,12)(17,33,50,61)(18,64,51,36)(19,35,52,63)(20,62,49,34)(25,56,55,26)(27,54,53,28)(29,48,59,41)(30,44,60,47)(31,46,57,43)(32,42,58,45) );
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)], [(1,15,21,11),(2,16,22,12),(3,13,23,9),(4,14,24,10),(5,56,38,26),(6,53,39,27),(7,54,40,28),(8,55,37,25),(17,30,50,60),(18,31,51,57),(19,32,52,58),(20,29,49,59),(33,47,61,44),(34,48,62,41),(35,45,63,42),(36,46,64,43)], [(1,42,37,59),(2,41,38,58),(3,44,39,57),(4,43,40,60),(5,32,22,48),(6,31,23,47),(7,30,24,46),(8,29,21,45),(9,61,53,51),(10,64,54,50),(11,63,55,49),(12,62,56,52),(13,33,27,18),(14,36,28,17),(15,35,25,20),(16,34,26,19)], [(1,2,21,22),(3,4,23,24),(5,37,38,8),(6,7,39,40),(9,14,13,10),(11,16,15,12),(17,33,50,61),(18,64,51,36),(19,35,52,63),(20,62,49,34),(25,56,55,26),(27,54,53,28),(29,48,59,41),(30,44,60,47),(31,46,57,43),(32,42,58,45)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2- 1+4 |
| kernel | C42.168D4 | C4×C4⋊C4 | C24.C22 | C24.3C22 | C23.67C23 | C23.Q8 | C2×C4×Q8 | C2×C4.4D4 | C2×C4⋊Q8 | C42 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42.168D4 ►in GL6(𝔽5)
| 2 | 4 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [2,3,0,0,0,0,4,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[1,4,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C42.168D4 in GAP, Magma, Sage, TeX
C_4^2._{168}D_4 % in TeX
G:=Group("C4^2.168D4"); // GroupNames label
G:=SmallGroup(128,1277);
// by ID
G=gap.SmallGroup(128,1277);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,184,675,80]);
// 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=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations