p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊47D4, C24.122C23, C23.756C24, C4⋊3(C4⋊D4), (C22×C4)⋊48D4, C42⋊9C4⋊40C2, C22⋊1(C4⋊1D4), (C22×C42)⋊17C2, C23.374(C2×D4), (C23×C4).683C22, C22.466(C22×D4), (C22×C4).1486C23, (C2×C42).1091C22, (C22×D4).313C22, C24.3C22⋊101C2, C2.56(C22.26C24), (C2×C4⋊D4)⋊42C2, (C2×C4⋊1D4)⋊12C2, (C2×C4).687(C2×D4), C2.49(C2×C4⋊D4), C2.16(C2×C4⋊1D4), (C2×C4).672(C4○D4), (C2×C4⋊C4).559C22, C22.597(C2×C4○D4), (C2×C22⋊C4).366C22, SmallGroup(128,1588)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊47D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 948 in 462 conjugacy classes, 144 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊1D4, C23×C4, C22×D4, C42⋊9C4, C24.3C22, C22×C42, C2×C4⋊D4, C2×C4⋊1D4, C42⋊47D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C4⋊1D4, C22×D4, C2×C4○D4, C2×C4⋊D4, C2×C4⋊1D4, C22.26C24, C42⋊47D4
►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 17 9 45)(2 18 10 46)(3 19 11 47)(4 20 12 48)(5 60 39 30)(6 57 40 31)(7 58 37 32)(8 59 38 29)(13 49 41 21)(14 50 42 22)(15 51 43 23)(16 52 44 24)(25 63 55 34)(26 64 56 35)(27 61 53 36)(28 62 54 33)
(1 49 53 6)(2 52 54 5)(3 51 55 8)(4 50 56 7)(9 21 27 40)(10 24 28 39)(11 23 25 38)(12 22 26 37)(13 36 31 17)(14 35 32 20)(15 34 29 19)(16 33 30 18)(41 61 57 45)(42 64 58 48)(43 63 59 47)(44 62 60 46)
(1 45)(2 48)(3 47)(4 46)(5 42)(6 41)(7 44)(8 43)(9 17)(10 20)(11 19)(12 18)(13 40)(14 39)(15 38)(16 37)(21 31)(22 30)(23 29)(24 32)(25 34)(26 33)(27 36)(28 35)(49 57)(50 60)(51 59)(52 58)(53 61)(54 64)(55 63)(56 62)
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), (1,17,9,45)(2,18,10,46)(3,19,11,47)(4,20,12,48)(5,60,39,30)(6,57,40,31)(7,58,37,32)(8,59,38,29)(13,49,41,21)(14,50,42,22)(15,51,43,23)(16,52,44,24)(25,63,55,34)(26,64,56,35)(27,61,53,36)(28,62,54,33), (1,49,53,6)(2,52,54,5)(3,51,55,8)(4,50,56,7)(9,21,27,40)(10,24,28,39)(11,23,25,38)(12,22,26,37)(13,36,31,17)(14,35,32,20)(15,34,29,19)(16,33,30,18)(41,61,57,45)(42,64,58,48)(43,63,59,47)(44,62,60,46), (1,45)(2,48)(3,47)(4,46)(5,42)(6,41)(7,44)(8,43)(9,17)(10,20)(11,19)(12,18)(13,40)(14,39)(15,38)(16,37)(21,31)(22,30)(23,29)(24,32)(25,34)(26,33)(27,36)(28,35)(49,57)(50,60)(51,59)(52,58)(53,61)(54,64)(55,63)(56,62)>;
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), (1,17,9,45)(2,18,10,46)(3,19,11,47)(4,20,12,48)(5,60,39,30)(6,57,40,31)(7,58,37,32)(8,59,38,29)(13,49,41,21)(14,50,42,22)(15,51,43,23)(16,52,44,24)(25,63,55,34)(26,64,56,35)(27,61,53,36)(28,62,54,33), (1,49,53,6)(2,52,54,5)(3,51,55,8)(4,50,56,7)(9,21,27,40)(10,24,28,39)(11,23,25,38)(12,22,26,37)(13,36,31,17)(14,35,32,20)(15,34,29,19)(16,33,30,18)(41,61,57,45)(42,64,58,48)(43,63,59,47)(44,62,60,46), (1,45)(2,48)(3,47)(4,46)(5,42)(6,41)(7,44)(8,43)(9,17)(10,20)(11,19)(12,18)(13,40)(14,39)(15,38)(16,37)(21,31)(22,30)(23,29)(24,32)(25,34)(26,33)(27,36)(28,35)(49,57)(50,60)(51,59)(52,58)(53,61)(54,64)(55,63)(56,62) );
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)], [(1,17,9,45),(2,18,10,46),(3,19,11,47),(4,20,12,48),(5,60,39,30),(6,57,40,31),(7,58,37,32),(8,59,38,29),(13,49,41,21),(14,50,42,22),(15,51,43,23),(16,52,44,24),(25,63,55,34),(26,64,56,35),(27,61,53,36),(28,62,54,33)], [(1,49,53,6),(2,52,54,5),(3,51,55,8),(4,50,56,7),(9,21,27,40),(10,24,28,39),(11,23,25,38),(12,22,26,37),(13,36,31,17),(14,35,32,20),(15,34,29,19),(16,33,30,18),(41,61,57,45),(42,64,58,48),(43,63,59,47),(44,62,60,46)], [(1,45),(2,48),(3,47),(4,46),(5,42),(6,41),(7,44),(8,43),(9,17),(10,20),(11,19),(12,18),(13,40),(14,39),(15,38),(16,37),(21,31),(22,30),(23,29),(24,32),(25,34),(26,33),(27,36),(28,35),(49,57),(50,60),(51,59),(52,58),(53,61),(54,64),(55,63),(56,62)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4X | 4Y | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 |
| kernel | C42⋊47D4 | C42⋊9C4 | C24.3C22 | C22×C42 | C2×C4⋊D4 | C2×C4⋊1D4 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 1 | 6 | 1 | 6 | 1 | 4 | 12 | 12 |
Matrix representation of C42⋊47D4 ►in GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,4,3],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,2],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,3,0,0,0,0,0,3],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,3,0,0,0,0,4,3] >;
C42⋊47D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{47}D_4 % in TeX
G:=Group("C4^2:47D4"); // GroupNames label
G:=SmallGroup(128,1588);
// by ID
G=gap.SmallGroup(128,1588);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,184,2019]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations