p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2.9(C4×D8), C4.66(C4×D4), D4⋊C4⋊4C4, C4⋊C4.211D4, (C2×C4).108D8, C2.3(C4⋊D8), C22.42(C2×D8), C22.162(C4×D4), C23.780(C2×D4), (C22×C4).695D4, C22.4Q16⋊16C2, C4.24(C4.4D4), C22.65(C4○D8), (C22×C8).48C22, C2.4(Q8.D4), C4.36(C42⋊C2), C4.29(C42⋊2C2), C22.84(C8⋊C22), (C2×C42).296C22, C2.5(C22.D8), (C22×D4).36C22, C2.14(SD16⋊C4), C22.124(C4⋊D4), C22.7C42⋊17C2, (C22×C4).1379C23, C2.6(C23.19D4), C22.73(C8.C22), C24.3C22.6C2, C2.10(C24.C22), C22.92(C22.D4), (C4×C4⋊C4)⋊6C2, (C2×C2.D8)⋊5C2, (C2×C8).41(C2×C4), C4⋊C4.149(C2×C4), (C2×D4).94(C2×C4), (C2×D4⋊C4).5C2, (C2×C4).1009(C2×D4), (C2×C4).575(C4○D4), (C2×C4⋊C4).773C22, (C2×C4).397(C22×C4), SmallGroup(128,657)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊C4⋊C4
G = < a,b,c,d | a4=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=bc2, cd=dc >
Subgroups: 340 in 145 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, D4⋊C4, C2.D8, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C22.4Q16, C4×C4⋊C4, C24.3C22, C2×D4⋊C4, C2×C2.D8, D4⋊C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C24.C22, C4×D8, SD16⋊C4, C4⋊D8, Q8.D4, C22.D8, C23.19D4, D4⋊C4⋊C4
►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 11)(2 10)(3 9)(4 12)(5 53)(6 56)(7 55)(8 54)(13 19)(14 18)(15 17)(16 20)(21 28)(22 27)(23 26)(24 25)(29 35)(30 34)(31 33)(32 36)(37 51)(38 50)(39 49)(40 52)(41 46)(42 45)(43 48)(44 47)(57 64)(58 63)(59 62)(60 61)
(1 34 16 27)(2 33 13 26)(3 36 14 25)(4 35 15 28)(5 47 60 39)(6 46 57 38)(7 45 58 37)(8 48 59 40)(9 31 18 23)(10 30 19 22)(11 29 20 21)(12 32 17 24)(41 63 50 55)(42 62 51 54)(43 61 52 53)(44 64 49 56)
(1 49 17 40)(2 50 18 37)(3 51 19 38)(4 52 20 39)(5 35 53 21)(6 36 54 22)(7 33 55 23)(8 34 56 24)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(25 62 30 57)(26 63 31 58)(27 64 32 59)(28 61 29 60)
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), (1,11)(2,10)(3,9)(4,12)(5,53)(6,56)(7,55)(8,54)(13,19)(14,18)(15,17)(16,20)(21,28)(22,27)(23,26)(24,25)(29,35)(30,34)(31,33)(32,36)(37,51)(38,50)(39,49)(40,52)(41,46)(42,45)(43,48)(44,47)(57,64)(58,63)(59,62)(60,61), (1,34,16,27)(2,33,13,26)(3,36,14,25)(4,35,15,28)(5,47,60,39)(6,46,57,38)(7,45,58,37)(8,48,59,40)(9,31,18,23)(10,30,19,22)(11,29,20,21)(12,32,17,24)(41,63,50,55)(42,62,51,54)(43,61,52,53)(44,64,49,56), (1,49,17,40)(2,50,18,37)(3,51,19,38)(4,52,20,39)(5,35,53,21)(6,36,54,22)(7,33,55,23)(8,34,56,24)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(25,62,30,57)(26,63,31,58)(27,64,32,59)(28,61,29,60)>;
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), (1,11)(2,10)(3,9)(4,12)(5,53)(6,56)(7,55)(8,54)(13,19)(14,18)(15,17)(16,20)(21,28)(22,27)(23,26)(24,25)(29,35)(30,34)(31,33)(32,36)(37,51)(38,50)(39,49)(40,52)(41,46)(42,45)(43,48)(44,47)(57,64)(58,63)(59,62)(60,61), (1,34,16,27)(2,33,13,26)(3,36,14,25)(4,35,15,28)(5,47,60,39)(6,46,57,38)(7,45,58,37)(8,48,59,40)(9,31,18,23)(10,30,19,22)(11,29,20,21)(12,32,17,24)(41,63,50,55)(42,62,51,54)(43,61,52,53)(44,64,49,56), (1,49,17,40)(2,50,18,37)(3,51,19,38)(4,52,20,39)(5,35,53,21)(6,36,54,22)(7,33,55,23)(8,34,56,24)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(25,62,30,57)(26,63,31,58)(27,64,32,59)(28,61,29,60) );
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)], [(1,11),(2,10),(3,9),(4,12),(5,53),(6,56),(7,55),(8,54),(13,19),(14,18),(15,17),(16,20),(21,28),(22,27),(23,26),(24,25),(29,35),(30,34),(31,33),(32,36),(37,51),(38,50),(39,49),(40,52),(41,46),(42,45),(43,48),(44,47),(57,64),(58,63),(59,62),(60,61)], [(1,34,16,27),(2,33,13,26),(3,36,14,25),(4,35,15,28),(5,47,60,39),(6,46,57,38),(7,45,58,37),(8,48,59,40),(9,31,18,23),(10,30,19,22),(11,29,20,21),(12,32,17,24),(41,63,50,55),(42,62,51,54),(43,61,52,53),(44,64,49,56)], [(1,49,17,40),(2,50,18,37),(3,51,19,38),(4,52,20,39),(5,35,53,21),(6,36,54,22),(7,33,55,23),(8,34,56,24),(9,45,13,41),(10,46,14,42),(11,47,15,43),(12,48,16,44),(25,62,30,57),(26,63,31,58),(27,64,32,59),(28,61,29,60)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | D4⋊C4⋊C4 | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C24.3C22 | C2×D4⋊C4 | C2×C2.D8 | D4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 1 | 1 |
Matrix representation of D4⋊C4⋊C4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 15 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 2 | 1 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,1,15,0,0,0,0,16],[16,0,0,0,0,0,3,3,0,0,0,3,14,0,0,0,0,0,4,0,0,0,0,0,4],[13,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,2,0,0,0,16,1] >;
D4⋊C4⋊C4 in GAP, Magma, Sage, TeX
D_4\rtimes C_4\rtimes C_4
% in TeX
G:=Group("D4:C4:C4"); // GroupNames label
G:=SmallGroup(128,657);
// by ID
G=gap.SmallGroup(128,657);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2804,718,172]);
// 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=b*c^2,c*d=d*c>;
// generators/relations