direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4×C4⋊C4, C23.230C24, C24.555C23, C22.482- 1+4, C22.662+ 1+4, C2.2D42, C4⋊5(C4×D4), C2.2(D4×Q8), (C4×D4)⋊21C4, C42⋊23(C2×C4), (C2×D4).35Q8, (C2×D4).341D4, C42⋊9C4⋊12C2, C2.3(D4⋊6D4), C2.2(D4⋊3Q8), C23.415(C2×D4), C23.113(C2×Q8), C23.7Q8⋊23C2, C23.8Q8⋊13C2, C22.38(C22×Q8), C23.129(C22×C4), (C23×C4).304C22, (C2×C42).429C22, C22.121(C23×C4), C22.105(C22×D4), (C22×C4).1245C23, (C22×D4).609C22, C23.65C23⋊22C2, C2.C42.58C22, C2.27(C23.33C23), C4⋊1(C2×C4⋊C4), (C4×C4⋊C4)⋊36C2, C2.31(C2×C4×D4), C4⋊C4⋊43(C2×C4), C22⋊1(C2×C4⋊C4), (C2×C4×D4).33C2, (C22×C4⋊C4)⋊9C2, C22⋊C4⋊40(C2×C4), (C22×C4)⋊31(C2×C4), C2.14(C22×C4⋊C4), (C2×C4).299(C2×Q8), (C2×D4).243(C2×C4), (C2×C4).1069(C2×D4), (C2×C4).793(C4○D4), (C2×C4⋊C4).820C22, (C2×C4).566(C22×C4), C22.115(C2×C4○D4), (C2×C22⋊C4).438C22, (C2×D4)○(C2×C4⋊C4), (C2×C4⋊C4)○(C22×D4), SmallGroup(128,1080)
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=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 636 in 388 conjugacy classes, 196 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, C4×C4⋊C4, C23.7Q8, C42⋊9C4, C23.8Q8, C23.65C23, C22×C4⋊C4, C2×C4×D4, C2×C4×D4, D4×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C4⋊C4, C2×C4×D4, C23.33C23, D42, D4⋊6D4, D4×Q8, D4⋊3Q8, 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 50)(2 49)(3 52)(4 51)(5 56)(6 55)(7 54)(8 53)(9 35)(10 34)(11 33)(12 36)(13 39)(14 38)(15 37)(16 40)(17 44)(18 43)(19 42)(20 41)(21 48)(22 47)(23 46)(24 45)(25 29)(26 32)(27 31)(28 30)(57 63)(58 62)(59 61)(60 64)
(1 13 51 40)(2 14 52 37)(3 15 49 38)(4 16 50 39)(5 44 55 20)(6 41 56 17)(7 42 53 18)(8 43 54 19)(9 31 36 28)(10 32 33 25)(11 29 34 26)(12 30 35 27)(21 61 45 60)(22 62 46 57)(23 63 47 58)(24 64 48 59)
(1 56 29 63)(2 53 30 64)(3 54 31 61)(4 55 32 62)(5 25 57 50)(6 26 58 51)(7 27 59 52)(8 28 60 49)(9 45 38 19)(10 46 39 20)(11 47 40 17)(12 48 37 18)(13 41 34 23)(14 42 35 24)(15 43 36 21)(16 44 33 22)
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), (1,50)(2,49)(3,52)(4,51)(5,56)(6,55)(7,54)(8,53)(9,35)(10,34)(11,33)(12,36)(13,39)(14,38)(15,37)(16,40)(17,44)(18,43)(19,42)(20,41)(21,48)(22,47)(23,46)(24,45)(25,29)(26,32)(27,31)(28,30)(57,63)(58,62)(59,61)(60,64), (1,13,51,40)(2,14,52,37)(3,15,49,38)(4,16,50,39)(5,44,55,20)(6,41,56,17)(7,42,53,18)(8,43,54,19)(9,31,36,28)(10,32,33,25)(11,29,34,26)(12,30,35,27)(21,61,45,60)(22,62,46,57)(23,63,47,58)(24,64,48,59), (1,56,29,63)(2,53,30,64)(3,54,31,61)(4,55,32,62)(5,25,57,50)(6,26,58,51)(7,27,59,52)(8,28,60,49)(9,45,38,19)(10,46,39,20)(11,47,40,17)(12,48,37,18)(13,41,34,23)(14,42,35,24)(15,43,36,21)(16,44,33,22)>;
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), (1,50)(2,49)(3,52)(4,51)(5,56)(6,55)(7,54)(8,53)(9,35)(10,34)(11,33)(12,36)(13,39)(14,38)(15,37)(16,40)(17,44)(18,43)(19,42)(20,41)(21,48)(22,47)(23,46)(24,45)(25,29)(26,32)(27,31)(28,30)(57,63)(58,62)(59,61)(60,64), (1,13,51,40)(2,14,52,37)(3,15,49,38)(4,16,50,39)(5,44,55,20)(6,41,56,17)(7,42,53,18)(8,43,54,19)(9,31,36,28)(10,32,33,25)(11,29,34,26)(12,30,35,27)(21,61,45,60)(22,62,46,57)(23,63,47,58)(24,64,48,59), (1,56,29,63)(2,53,30,64)(3,54,31,61)(4,55,32,62)(5,25,57,50)(6,26,58,51)(7,27,59,52)(8,28,60,49)(9,45,38,19)(10,46,39,20)(11,47,40,17)(12,48,37,18)(13,41,34,23)(14,42,35,24)(15,43,36,21)(16,44,33,22) );
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)], [(1,50),(2,49),(3,52),(4,51),(5,56),(6,55),(7,54),(8,53),(9,35),(10,34),(11,33),(12,36),(13,39),(14,38),(15,37),(16,40),(17,44),(18,43),(19,42),(20,41),(21,48),(22,47),(23,46),(24,45),(25,29),(26,32),(27,31),(28,30),(57,63),(58,62),(59,61),(60,64)], [(1,13,51,40),(2,14,52,37),(3,15,49,38),(4,16,50,39),(5,44,55,20),(6,41,56,17),(7,42,53,18),(8,43,54,19),(9,31,36,28),(10,32,33,25),(11,29,34,26),(12,30,35,27),(21,61,45,60),(22,62,46,57),(23,63,47,58),(24,64,48,59)], [(1,56,29,63),(2,53,30,64),(3,54,31,61),(4,55,32,62),(5,25,57,50),(6,26,58,51),(7,27,59,52),(8,28,60,49),(9,45,38,19),(10,46,39,20),(11,47,40,17),(12,48,37,18),(13,41,34,23),(14,42,35,24),(15,43,36,21),(16,44,33,22)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4P | 4Q | ··· | 4AH |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 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 | C4 | D4 | D4 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | D4×C4⋊C4 | C4×C4⋊C4 | C23.7Q8 | C42⋊9C4 | C23.8Q8 | C23.65C23 | C22×C4⋊C4 | C2×C4×D4 | C4×D4 | C4⋊C4 | C2×D4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 3 | 16 | 4 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of D4×C4⋊C4 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,4,0,0,0,2,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,2,1],[4,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4] >;
D4×C4⋊C4 in GAP, Magma, Sage, TeX
D_4\times C_4\rtimes C_4
% in TeX
G:=Group("D4xC4:C4"); // GroupNames label
G:=SmallGroup(128,1080);
// by ID
G=gap.SmallGroup(128,1080);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,184,346]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations