p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.166D4, C23.438C24, C24.321C23, C22.2272+ 1+4, C22.1752- 1+4, (C2×Q8)⋊25D4, (C2×D4)⋊15Q8, C2.21(D4×Q8), C4⋊5(C22⋊Q8), C42⋊9C4⋊27C2, C23.20(C2×Q8), C4.167(C4⋊D4), C2.29(D4⋊3Q8), C2.22(Q8⋊6D4), (C22×C4).95C23, C23.Q8⋊27C2, C23.7Q8⋊66C2, C22.96(C22×Q8), (C23×C4).391C22, (C2×C42).544C22, C22.289(C22×D4), (C22×D4).528C22, (C22×Q8).432C22, C23.65C23⋊84C2, C24.3C22.43C2, C2.C42.181C22, C2.36(C22.50C24), C2.13(C22.31C24), (C2×C4×Q8)⋊22C2, (C2×C4×D4).59C2, (C2×C4).70(C2×D4), C2.33(C2×C4⋊D4), (C2×C22⋊Q8)⋊20C2, (C2×C4).308(C2×Q8), C2.29(C2×C22⋊Q8), (C2×C4).820(C4○D4), (C2×C4⋊C4).298C22, C22.315(C2×C4○D4), (C2×C22⋊C4).174C22, SmallGroup(128,1270)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.166D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 564 in 310 conjugacy classes, 124 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C23.7Q8, C42⋊9C4, C23.65C23, C24.3C22, C23.Q8, C2×C4×D4, C2×C4×Q8, C2×C22⋊Q8, C42.166D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C4⋊D4, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4⋊D4, C2×C22⋊Q8, C22.31C24, D4×Q8, Q8⋊6D4, D4⋊3Q8, C22.50C24, C42.166D4
►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 11 20)(2 16 12 17)(3 13 9 18)(4 14 10 19)(5 29 28 63)(6 30 25 64)(7 31 26 61)(8 32 27 62)(21 44 45 52)(22 41 46 49)(23 42 47 50)(24 43 48 51)(33 38 55 59)(34 39 56 60)(35 40 53 57)(36 37 54 58)
(1 62 52 39)(2 61 49 38)(3 64 50 37)(4 63 51 40)(5 48 53 19)(6 47 54 18)(7 46 55 17)(8 45 56 20)(9 30 42 58)(10 29 43 57)(11 32 44 60)(12 31 41 59)(13 25 23 36)(14 28 24 35)(15 27 21 34)(16 26 22 33)
(1 55 11 33)(2 34 12 56)(3 53 9 35)(4 36 10 54)(5 42 28 50)(6 51 25 43)(7 44 26 52)(8 49 27 41)(13 40 18 57)(14 58 19 37)(15 38 20 59)(16 60 17 39)(21 61 45 31)(22 32 46 62)(23 63 47 29)(24 30 48 64)
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), (1,15,11,20)(2,16,12,17)(3,13,9,18)(4,14,10,19)(5,29,28,63)(6,30,25,64)(7,31,26,61)(8,32,27,62)(21,44,45,52)(22,41,46,49)(23,42,47,50)(24,43,48,51)(33,38,55,59)(34,39,56,60)(35,40,53,57)(36,37,54,58), (1,62,52,39)(2,61,49,38)(3,64,50,37)(4,63,51,40)(5,48,53,19)(6,47,54,18)(7,46,55,17)(8,45,56,20)(9,30,42,58)(10,29,43,57)(11,32,44,60)(12,31,41,59)(13,25,23,36)(14,28,24,35)(15,27,21,34)(16,26,22,33), (1,55,11,33)(2,34,12,56)(3,53,9,35)(4,36,10,54)(5,42,28,50)(6,51,25,43)(7,44,26,52)(8,49,27,41)(13,40,18,57)(14,58,19,37)(15,38,20,59)(16,60,17,39)(21,61,45,31)(22,32,46,62)(23,63,47,29)(24,30,48,64)>;
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), (1,15,11,20)(2,16,12,17)(3,13,9,18)(4,14,10,19)(5,29,28,63)(6,30,25,64)(7,31,26,61)(8,32,27,62)(21,44,45,52)(22,41,46,49)(23,42,47,50)(24,43,48,51)(33,38,55,59)(34,39,56,60)(35,40,53,57)(36,37,54,58), (1,62,52,39)(2,61,49,38)(3,64,50,37)(4,63,51,40)(5,48,53,19)(6,47,54,18)(7,46,55,17)(8,45,56,20)(9,30,42,58)(10,29,43,57)(11,32,44,60)(12,31,41,59)(13,25,23,36)(14,28,24,35)(15,27,21,34)(16,26,22,33), (1,55,11,33)(2,34,12,56)(3,53,9,35)(4,36,10,54)(5,42,28,50)(6,51,25,43)(7,44,26,52)(8,49,27,41)(13,40,18,57)(14,58,19,37)(15,38,20,59)(16,60,17,39)(21,61,45,31)(22,32,46,62)(23,63,47,29)(24,30,48,64) );
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)], [(1,15,11,20),(2,16,12,17),(3,13,9,18),(4,14,10,19),(5,29,28,63),(6,30,25,64),(7,31,26,61),(8,32,27,62),(21,44,45,52),(22,41,46,49),(23,42,47,50),(24,43,48,51),(33,38,55,59),(34,39,56,60),(35,40,53,57),(36,37,54,58)], [(1,62,52,39),(2,61,49,38),(3,64,50,37),(4,63,51,40),(5,48,53,19),(6,47,54,18),(7,46,55,17),(8,45,56,20),(9,30,42,58),(10,29,43,57),(11,32,44,60),(12,31,41,59),(13,25,23,36),(14,28,24,35),(15,27,21,34),(16,26,22,33)], [(1,55,11,33),(2,34,12,56),(3,53,9,35),(4,36,10,54),(5,42,28,50),(6,51,25,43),(7,44,26,52),(8,49,27,41),(13,40,18,57),(14,58,19,37),(15,38,20,59),(16,60,17,39),(21,61,45,31),(22,32,46,62),(23,63,47,29),(24,30,48,64)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.166D4 | C23.7Q8 | C42⋊9C4 | C23.65C23 | C24.3C22 | C23.Q8 | C2×C4×D4 | C2×C4×Q8 | C2×C22⋊Q8 | C42 | C2×D4 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 1 | 1 |
Matrix representation of C42.166D4 ►in GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 2 | 3 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,3,0,0,0,0,0,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,1,3] >;
C42.166D4 in GAP, Magma, Sage, TeX
C_4^2._{166}D_4 % in TeX
G:=Group("C4^2.166D4"); // GroupNames label
G:=SmallGroup(128,1270);
// by ID
G=gap.SmallGroup(128,1270);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,675,80]);
// 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*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations