direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C4.4D4, C43⋊10C2, C42.343D4, C24.188C23, C23.185C24, C4.35(C4×D4), C42⋊38(C2×C4), C4○3(C42⋊8C4), C42⋊8C4⋊78C2, C23.7(C22×C4), C22.76(C23×C4), (C23×C4).37C22, C22.81(C22×D4), (C22×C4).749C23, C4○4(C24.3C22), C4○6(C24.C22), (C2×C42).1008C22, (C22×D4).470C22, (C22×Q8).393C22, C4○4(C23.67C23), C24.C22⋊192C2, C24.3C22.84C2, C2.5(C22.26C24), C23.67C23⋊114C2, C2.C42.468C22, C2.9(C23.36C23), (C2×C4×Q8)⋊3C2, C2.17(C2×C4×D4), (C2×C4×D4).28C2, (C2×Q8)⋊23(C2×C4), C2.14(C4×C4○D4), C22⋊C4⋊27(C2×C4), (C4×C22⋊C4)⋊30C2, (C2×C4).824(C2×D4), C2.5(C2×C4.4D4), (C2×D4).166(C2×C4), C22.77(C2×C4○D4), (C2×C4)○2(C42⋊8C4), (C2×C4).639(C4○D4), (C2×C4⋊C4).799C22, (C2×C4).289(C22×C4), (C2×C4.4D4).45C2, (C22×C4)○(C42⋊8C4), (C2×C22⋊C4).480C22, (C2×C4)○3(C24.C22), (C2×C4)○2(C24.3C22), (C2×C4)○2(C23.67C23), (C22×C4)○(C23.67C23), (C2×C4)○(C2×C4.4D4), (C22×C4)○(C2×C4.4D4), SmallGroup(128,1035)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C4.4D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 540 in 326 conjugacy classes, 160 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C4.4D4, C23×C4, C22×D4, C22×Q8, C43, C4×C22⋊C4, C42⋊8C4, C24.C22, C24.3C22, C23.67C23, C2×C4×D4, C2×C4×Q8, C2×C4.4D4, C4×C4.4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C4.4D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C4×C4○D4, C2×C4.4D4, C23.36C23, C22.26C24, C4×C4.4D4
►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 52 8 15)(2 49 5 16)(3 50 6 13)(4 51 7 14)(9 60 21 61)(10 57 22 62)(11 58 23 63)(12 59 24 64)(17 48 38 43)(18 45 39 44)(19 46 40 41)(20 47 37 42)(25 56 30 35)(26 53 31 36)(27 54 32 33)(28 55 29 34)
(1 55 23 44)(2 56 24 41)(3 53 21 42)(4 54 22 43)(5 35 12 46)(6 36 9 47)(7 33 10 48)(8 34 11 45)(13 26 60 37)(14 27 57 38)(15 28 58 39)(16 25 59 40)(17 51 32 62)(18 52 29 63)(19 49 30 64)(20 50 31 61)
(1 20 8 37)(2 17 5 38)(3 18 6 39)(4 19 7 40)(9 28 21 29)(10 25 22 30)(11 26 23 31)(12 27 24 32)(13 45 50 44)(14 46 51 41)(15 47 52 42)(16 48 49 43)(33 64 54 59)(34 61 55 60)(35 62 56 57)(36 63 53 58)
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), (1,52,8,15)(2,49,5,16)(3,50,6,13)(4,51,7,14)(9,60,21,61)(10,57,22,62)(11,58,23,63)(12,59,24,64)(17,48,38,43)(18,45,39,44)(19,46,40,41)(20,47,37,42)(25,56,30,35)(26,53,31,36)(27,54,32,33)(28,55,29,34), (1,55,23,44)(2,56,24,41)(3,53,21,42)(4,54,22,43)(5,35,12,46)(6,36,9,47)(7,33,10,48)(8,34,11,45)(13,26,60,37)(14,27,57,38)(15,28,58,39)(16,25,59,40)(17,51,32,62)(18,52,29,63)(19,49,30,64)(20,50,31,61), (1,20,8,37)(2,17,5,38)(3,18,6,39)(4,19,7,40)(9,28,21,29)(10,25,22,30)(11,26,23,31)(12,27,24,32)(13,45,50,44)(14,46,51,41)(15,47,52,42)(16,48,49,43)(33,64,54,59)(34,61,55,60)(35,62,56,57)(36,63,53,58)>;
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), (1,52,8,15)(2,49,5,16)(3,50,6,13)(4,51,7,14)(9,60,21,61)(10,57,22,62)(11,58,23,63)(12,59,24,64)(17,48,38,43)(18,45,39,44)(19,46,40,41)(20,47,37,42)(25,56,30,35)(26,53,31,36)(27,54,32,33)(28,55,29,34), (1,55,23,44)(2,56,24,41)(3,53,21,42)(4,54,22,43)(5,35,12,46)(6,36,9,47)(7,33,10,48)(8,34,11,45)(13,26,60,37)(14,27,57,38)(15,28,58,39)(16,25,59,40)(17,51,32,62)(18,52,29,63)(19,49,30,64)(20,50,31,61), (1,20,8,37)(2,17,5,38)(3,18,6,39)(4,19,7,40)(9,28,21,29)(10,25,22,30)(11,26,23,31)(12,27,24,32)(13,45,50,44)(14,46,51,41)(15,47,52,42)(16,48,49,43)(33,64,54,59)(34,61,55,60)(35,62,56,57)(36,63,53,58) );
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)], [(1,52,8,15),(2,49,5,16),(3,50,6,13),(4,51,7,14),(9,60,21,61),(10,57,22,62),(11,58,23,63),(12,59,24,64),(17,48,38,43),(18,45,39,44),(19,46,40,41),(20,47,37,42),(25,56,30,35),(26,53,31,36),(27,54,32,33),(28,55,29,34)], [(1,55,23,44),(2,56,24,41),(3,53,21,42),(4,54,22,43),(5,35,12,46),(6,36,9,47),(7,33,10,48),(8,34,11,45),(13,26,60,37),(14,27,57,38),(15,28,58,39),(16,25,59,40),(17,51,32,62),(18,52,29,63),(19,49,30,64),(20,50,31,61)], [(1,20,8,37),(2,17,5,38),(3,18,6,39),(4,19,7,40),(9,28,21,29),(10,25,22,30),(11,26,23,31),(12,27,24,32),(13,45,50,44),(14,46,51,41),(15,47,52,42),(16,48,49,43),(33,64,54,59),(34,61,55,60),(35,62,56,57),(36,63,53,58)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AF | 4AG | ··· | 4AR |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 |
| kernel | C4×C4.4D4 | C43 | C4×C22⋊C4 | C42⋊8C4 | C24.C22 | C24.3C22 | C23.67C23 | C2×C4×D4 | C2×C4×Q8 | C2×C4.4D4 | C4.4D4 | C42 | C2×C4 |
| # reps | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 16 | 4 | 20 |
Matrix representation of C4×C4.4D4 ►in GL5(𝔽5)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(5))| [3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
C4×C4.4D4 in GAP, Magma, Sage, TeX
C_4\times C_4._4D_4
% in TeX
G:=Group("C4xC4.4D4"); // GroupNames label
G:=SmallGroup(128,1035);
// by ID
G=gap.SmallGroup(128,1035);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations