direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×D4⋊C4, D4⋊2C42, C42.423D4, C2.2(C4×D8), (C4×D4)⋊11C4, (C2×C4).166D8, C4.108(C4×D4), C4.3(C2×C42), C2.2(C4×SD16), C22.86(C4×D4), C22.28(C2×D8), C42.260(C2×C4), (C2×C4).128SD16, (C22×C4).812D4, C23.730(C2×D4), C4○2(C22.4Q16), C22.4Q16⋊51C2, C4.3(C42⋊C2), C22.36(C4○D8), C22.41(C2×SD16), (C22×C8).472C22, (C2×C42).1046C22, (C22×C4).1307C23, C2.3(C23.24D4), (C22×D4).446C22, (C2×C4×C8)⋊8C2, (C4×C4⋊C4)⋊3C2, C4⋊C4⋊22(C2×C4), (C2×C8)⋊28(C2×C4), (C2×C4×D4).11C2, C2.3(C2×D4⋊C4), C2.18(C4×C22⋊C4), (C2×D4).198(C2×C4), (C2×C4).1302(C2×D4), (C2×D4⋊C4).39C2, (C2×C4).538(C4○D4), (C2×C4⋊C4).746C22, (C2×C4).350(C22×C4), (C2×C4).400(C22⋊C4), C22.119(C2×C22⋊C4), SmallGroup(128,492)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×D4⋊C4
G = < a,b,c,d | a4=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 396 in 198 conjugacy classes, 92 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, C4×C8, 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×C4×C8, C2×D4⋊C4, C2×C4×D4, C4×D4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4○D4, D4⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C2×D8, C2×SD16, C4○D8, C4×C22⋊C4, C2×D4⋊C4, C23.24D4, C4×D8, C4×SD16, C4×D4⋊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 18 51 41)(2 19 52 42)(3 20 49 43)(4 17 50 44)(5 38 55 64)(6 39 56 61)(7 40 53 62)(8 37 54 63)(9 13 46 22)(10 14 47 23)(11 15 48 24)(12 16 45 21)(25 29 60 33)(26 30 57 34)(27 31 58 35)(28 32 59 36)
(1 23)(2 24)(3 21)(4 22)(5 32)(6 29)(7 30)(8 31)(9 44)(10 41)(11 42)(12 43)(13 50)(14 51)(15 52)(16 49)(17 46)(18 47)(19 48)(20 45)(25 39)(26 40)(27 37)(28 38)(33 56)(34 53)(35 54)(36 55)(57 62)(58 63)(59 64)(60 61)
(1 34 23 40)(2 35 24 37)(3 36 21 38)(4 33 22 39)(5 20 59 12)(6 17 60 9)(7 18 57 10)(8 19 58 11)(13 61 50 29)(14 62 51 30)(15 63 52 31)(16 64 49 32)(25 46 56 44)(26 47 53 41)(27 48 54 42)(28 45 55 43)
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), (1,18,51,41)(2,19,52,42)(3,20,49,43)(4,17,50,44)(5,38,55,64)(6,39,56,61)(7,40,53,62)(8,37,54,63)(9,13,46,22)(10,14,47,23)(11,15,48,24)(12,16,45,21)(25,29,60,33)(26,30,57,34)(27,31,58,35)(28,32,59,36), (1,23)(2,24)(3,21)(4,22)(5,32)(6,29)(7,30)(8,31)(9,44)(10,41)(11,42)(12,43)(13,50)(14,51)(15,52)(16,49)(17,46)(18,47)(19,48)(20,45)(25,39)(26,40)(27,37)(28,38)(33,56)(34,53)(35,54)(36,55)(57,62)(58,63)(59,64)(60,61), (1,34,23,40)(2,35,24,37)(3,36,21,38)(4,33,22,39)(5,20,59,12)(6,17,60,9)(7,18,57,10)(8,19,58,11)(13,61,50,29)(14,62,51,30)(15,63,52,31)(16,64,49,32)(25,46,56,44)(26,47,53,41)(27,48,54,42)(28,45,55,43)>;
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), (1,18,51,41)(2,19,52,42)(3,20,49,43)(4,17,50,44)(5,38,55,64)(6,39,56,61)(7,40,53,62)(8,37,54,63)(9,13,46,22)(10,14,47,23)(11,15,48,24)(12,16,45,21)(25,29,60,33)(26,30,57,34)(27,31,58,35)(28,32,59,36), (1,23)(2,24)(3,21)(4,22)(5,32)(6,29)(7,30)(8,31)(9,44)(10,41)(11,42)(12,43)(13,50)(14,51)(15,52)(16,49)(17,46)(18,47)(19,48)(20,45)(25,39)(26,40)(27,37)(28,38)(33,56)(34,53)(35,54)(36,55)(57,62)(58,63)(59,64)(60,61), (1,34,23,40)(2,35,24,37)(3,36,21,38)(4,33,22,39)(5,20,59,12)(6,17,60,9)(7,18,57,10)(8,19,58,11)(13,61,50,29)(14,62,51,30)(15,63,52,31)(16,64,49,32)(25,46,56,44)(26,47,53,41)(27,48,54,42)(28,45,55,43) );
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)], [(1,18,51,41),(2,19,52,42),(3,20,49,43),(4,17,50,44),(5,38,55,64),(6,39,56,61),(7,40,53,62),(8,37,54,63),(9,13,46,22),(10,14,47,23),(11,15,48,24),(12,16,45,21),(25,29,60,33),(26,30,57,34),(27,31,58,35),(28,32,59,36)], [(1,23),(2,24),(3,21),(4,22),(5,32),(6,29),(7,30),(8,31),(9,44),(10,41),(11,42),(12,43),(13,50),(14,51),(15,52),(16,49),(17,46),(18,47),(19,48),(20,45),(25,39),(26,40),(27,37),(28,38),(33,56),(34,53),(35,54),(36,55),(57,62),(58,63),(59,64),(60,61)], [(1,34,23,40),(2,35,24,37),(3,36,21,38),(4,33,22,39),(5,20,59,12),(6,17,60,9),(7,18,57,10),(8,19,58,11),(13,61,50,29),(14,62,51,30),(15,63,52,31),(16,64,49,32),(25,46,56,44),(26,47,53,41),(27,48,54,42),(28,45,55,43)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D8 | SD16 | C4○D4 | C4○D8 |
| kernel | C4×D4⋊C4 | C22.4Q16 | C4×C4⋊C4 | C2×C4×C8 | C2×D4⋊C4 | C2×C4×D4 | D4⋊C4 | C4×D4 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 8 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C4×D4⋊C4 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 14 | 14 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,16,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,13,0,0,0,0,3,14,0,0,14,14] >;
C4×D4⋊C4 in GAP, Magma, Sage, TeX
C_4\times D_4\rtimes C_4
% in TeX
G:=Group("C4xD4:C4"); // GroupNames label
G:=SmallGroup(128,492);
// by ID
G=gap.SmallGroup(128,492);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations