p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊3C42, C42.93D4, (C4×D4)⋊12C4, D4⋊C4⋊9C4, C4.110(C4×D4), C4.5(C2×C42), C22.88(C4×D4), C2.2(D8⋊C4), C42.128(C2×C4), (C22×C4).671D4, C23.732(C2×D4), C22.4Q16⋊45C2, C4.5(C42⋊C2), C2.2(SD16⋊C4), C22.54(C8⋊C22), (C2×C42).235C22, (C22×C8).380C22, (C22×C4).1309C23, C2.3(C23.37D4), C2.3(C23.36D4), (C22×D4).447C22, C22.43(C8.C22), (C4×C4⋊C4)⋊4C2, C4⋊C4⋊23(C2×C4), (C2×C8)⋊24(C2×C4), (C2×C4×D4).12C2, (C2×C8⋊C4)⋊21C2, C2.20(C4×C22⋊C4), (C2×D4).199(C2×C4), (C2×C4).1304(C2×D4), (C2×D4⋊C4).32C2, (C2×C4).540(C4○D4), (C2×C4⋊C4).748C22, (C2×C4).352(C22×C4), (C2×C4).327(C22⋊C4), C22.121(C2×C22⋊C4), SmallGroup(128,494)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊C42
G = < a,b,c,d | a4=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=a2b, cd=dc >
Subgroups: 396 in 194 conjugacy classes, 84 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C8⋊C4, D4⋊C4, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C22×C8, C23×C4, C22×D4, C22.4Q16, C4×C4⋊C4, C2×C8⋊C4, C2×D4⋊C4, C2×C4×D4, D4⋊C42
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⋊C22, C8.C22, C4×C22⋊C4, C23.36D4, C23.37D4, SD16⋊C4, D8⋊C4, D4⋊C42
►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 7)(2 6)(3 5)(4 8)(9 61)(10 64)(11 63)(12 62)(13 18)(14 17)(15 20)(16 19)(21 25)(22 28)(23 27)(24 26)(29 34)(30 33)(31 36)(32 35)(37 42)(38 41)(39 44)(40 43)(45 51)(46 50)(47 49)(48 52)(53 58)(54 57)(55 60)(56 59)
(1 51 16 39)(2 50 13 38)(3 49 14 37)(4 52 15 40)(5 46 17 41)(6 45 18 44)(7 48 19 43)(8 47 20 42)(9 32 55 27)(10 31 56 26)(11 30 53 25)(12 29 54 28)(21 62 33 57)(22 61 34 60)(23 64 35 59)(24 63 36 58)
(1 21 6 26)(2 22 7 27)(3 23 8 28)(4 24 5 25)(9 50 61 48)(10 51 62 45)(11 52 63 46)(12 49 64 47)(13 34 19 32)(14 35 20 29)(15 36 17 30)(16 33 18 31)(37 59 42 54)(38 60 43 55)(39 57 44 56)(40 58 41 53)
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), (1,7)(2,6)(3,5)(4,8)(9,61)(10,64)(11,63)(12,62)(13,18)(14,17)(15,20)(16,19)(21,25)(22,28)(23,27)(24,26)(29,34)(30,33)(31,36)(32,35)(37,42)(38,41)(39,44)(40,43)(45,51)(46,50)(47,49)(48,52)(53,58)(54,57)(55,60)(56,59), (1,51,16,39)(2,50,13,38)(3,49,14,37)(4,52,15,40)(5,46,17,41)(6,45,18,44)(7,48,19,43)(8,47,20,42)(9,32,55,27)(10,31,56,26)(11,30,53,25)(12,29,54,28)(21,62,33,57)(22,61,34,60)(23,64,35,59)(24,63,36,58), (1,21,6,26)(2,22,7,27)(3,23,8,28)(4,24,5,25)(9,50,61,48)(10,51,62,45)(11,52,63,46)(12,49,64,47)(13,34,19,32)(14,35,20,29)(15,36,17,30)(16,33,18,31)(37,59,42,54)(38,60,43,55)(39,57,44,56)(40,58,41,53)>;
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), (1,7)(2,6)(3,5)(4,8)(9,61)(10,64)(11,63)(12,62)(13,18)(14,17)(15,20)(16,19)(21,25)(22,28)(23,27)(24,26)(29,34)(30,33)(31,36)(32,35)(37,42)(38,41)(39,44)(40,43)(45,51)(46,50)(47,49)(48,52)(53,58)(54,57)(55,60)(56,59), (1,51,16,39)(2,50,13,38)(3,49,14,37)(4,52,15,40)(5,46,17,41)(6,45,18,44)(7,48,19,43)(8,47,20,42)(9,32,55,27)(10,31,56,26)(11,30,53,25)(12,29,54,28)(21,62,33,57)(22,61,34,60)(23,64,35,59)(24,63,36,58), (1,21,6,26)(2,22,7,27)(3,23,8,28)(4,24,5,25)(9,50,61,48)(10,51,62,45)(11,52,63,46)(12,49,64,47)(13,34,19,32)(14,35,20,29)(15,36,17,30)(16,33,18,31)(37,59,42,54)(38,60,43,55)(39,57,44,56)(40,58,41,53) );
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)], [(1,7),(2,6),(3,5),(4,8),(9,61),(10,64),(11,63),(12,62),(13,18),(14,17),(15,20),(16,19),(21,25),(22,28),(23,27),(24,26),(29,34),(30,33),(31,36),(32,35),(37,42),(38,41),(39,44),(40,43),(45,51),(46,50),(47,49),(48,52),(53,58),(54,57),(55,60),(56,59)], [(1,51,16,39),(2,50,13,38),(3,49,14,37),(4,52,15,40),(5,46,17,41),(6,45,18,44),(7,48,19,43),(8,47,20,42),(9,32,55,27),(10,31,56,26),(11,30,53,25),(12,29,54,28),(21,62,33,57),(22,61,34,60),(23,64,35,59),(24,63,36,58)], [(1,21,6,26),(2,22,7,27),(3,23,8,28),(4,24,5,25),(9,50,61,48),(10,51,62,45),(11,52,63,46),(12,49,64,47),(13,34,19,32),(14,35,20,29),(15,36,17,30),(16,33,18,31),(37,59,42,54),(38,60,43,55),(39,57,44,56),(40,58,41,53)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | ··· | 4X | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | D4⋊C42 | C22.4Q16 | C4×C4⋊C4 | C2×C8⋊C4 | C2×D4⋊C4 | C2×C4×D4 | D4⋊C4 | C4×D4 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 8 | 2 | 2 | 4 | 3 | 1 |
Matrix representation of D4⋊C42 ►in GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 8 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 16 | 16 |
| 0 | 0 | 0 | 0 | 13 | 4 | 16 | 1 |
| 0 | 0 | 0 | 0 | 8 | 8 | 4 | 4 |
| 0 | 0 | 0 | 0 | 8 | 9 | 4 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 15 |
| 0 | 0 | 0 | 0 | 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 | 13 | 0 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[16,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[8,10,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,13,8,8,0,0,0,0,13,4,8,9,0,0,0,0,16,16,4,4,0,0,0,0,16,1,4,13],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,13,0,16,0,0,0,0,0,0,2,0,13,0,0,0,0,15,0,4,0] >;
D4⋊C42 in GAP, Magma, Sage, TeX
D_4\rtimes C_4^2
% in TeX
G:=Group("D4:C4^2"); // GroupNames label
G:=SmallGroup(128,494);
// by ID
G=gap.SmallGroup(128,494);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,2019,248,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,d*b*d^-1=a^2*b,c*d=d*c>;
// generators/relations