p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊20D4, C23.441C24, C24.322C23, C22.2302+ 1+4, C2.29D42, C4⋊C4⋊26D4, (C2×D4)⋊35D4, C4⋊2(C4⋊D4), C4⋊2(C4⋊1D4), C23.48(C2×D4), C23⋊2D4⋊19C2, C2.23(Q8⋊6D4), (C23×C4).394C22, (C2×C42).546C22, (C22×C4).834C23, C22.292(C22×D4), C24.3C22⋊54C2, (C22×D4).163C22, C2.21(C22.29C24), C2.C42.547C22, (C2×C4×D4)⋊45C2, (C4×C4⋊C4)⋊86C2, (C2×C4⋊1D4)⋊6C2, (C2×C4).72(C2×D4), (C2×C4⋊D4)⋊18C2, C2.11(C2×C4⋊1D4), C2.34(C2×C4⋊D4), (C2×C4).893(C4○D4), (C2×C4⋊C4).866C22, C22.318(C2×C4○D4), (C2×C22⋊C4).176C22, SmallGroup(128,1273)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊20D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1076 in 474 conjugacy classes, 132 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C4⋊1D4, C23×C4, C22×D4, C22×D4, C4×C4⋊C4, C24.3C22, C23⋊2D4, C2×C4×D4, C2×C4⋊D4, C2×C4⋊1D4, C2×C4⋊1D4, C42⋊20D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C4⋊1D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4⋊D4, C2×C4⋊1D4, C22.29C24, D42, Q8⋊6D4, C42⋊20D4
►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 15 10 50)(2 16 11 51)(3 13 12 52)(4 14 9 49)(5 63 37 36)(6 64 38 33)(7 61 39 34)(8 62 40 35)(17 44 46 22)(18 41 47 23)(19 42 48 24)(20 43 45 21)(25 57 54 30)(26 58 55 31)(27 59 56 32)(28 60 53 29)
(1 46 26 35)(2 18 27 63)(3 48 28 33)(4 20 25 61)(5 16 23 59)(6 52 24 29)(7 14 21 57)(8 50 22 31)(9 45 54 34)(10 17 55 62)(11 47 56 36)(12 19 53 64)(13 42 60 38)(15 44 58 40)(30 39 49 43)(32 37 51 41)
(1 11)(2 10)(3 9)(4 12)(5 44)(6 43)(7 42)(8 41)(13 14)(15 16)(17 63)(18 62)(19 61)(20 64)(21 38)(22 37)(23 40)(24 39)(25 53)(26 56)(27 55)(28 54)(29 30)(31 32)(33 45)(34 48)(35 47)(36 46)(49 52)(50 51)(57 60)(58 59)
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), (1,15,10,50)(2,16,11,51)(3,13,12,52)(4,14,9,49)(5,63,37,36)(6,64,38,33)(7,61,39,34)(8,62,40,35)(17,44,46,22)(18,41,47,23)(19,42,48,24)(20,43,45,21)(25,57,54,30)(26,58,55,31)(27,59,56,32)(28,60,53,29), (1,46,26,35)(2,18,27,63)(3,48,28,33)(4,20,25,61)(5,16,23,59)(6,52,24,29)(7,14,21,57)(8,50,22,31)(9,45,54,34)(10,17,55,62)(11,47,56,36)(12,19,53,64)(13,42,60,38)(15,44,58,40)(30,39,49,43)(32,37,51,41), (1,11)(2,10)(3,9)(4,12)(5,44)(6,43)(7,42)(8,41)(13,14)(15,16)(17,63)(18,62)(19,61)(20,64)(21,38)(22,37)(23,40)(24,39)(25,53)(26,56)(27,55)(28,54)(29,30)(31,32)(33,45)(34,48)(35,47)(36,46)(49,52)(50,51)(57,60)(58,59)>;
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), (1,15,10,50)(2,16,11,51)(3,13,12,52)(4,14,9,49)(5,63,37,36)(6,64,38,33)(7,61,39,34)(8,62,40,35)(17,44,46,22)(18,41,47,23)(19,42,48,24)(20,43,45,21)(25,57,54,30)(26,58,55,31)(27,59,56,32)(28,60,53,29), (1,46,26,35)(2,18,27,63)(3,48,28,33)(4,20,25,61)(5,16,23,59)(6,52,24,29)(7,14,21,57)(8,50,22,31)(9,45,54,34)(10,17,55,62)(11,47,56,36)(12,19,53,64)(13,42,60,38)(15,44,58,40)(30,39,49,43)(32,37,51,41), (1,11)(2,10)(3,9)(4,12)(5,44)(6,43)(7,42)(8,41)(13,14)(15,16)(17,63)(18,62)(19,61)(20,64)(21,38)(22,37)(23,40)(24,39)(25,53)(26,56)(27,55)(28,54)(29,30)(31,32)(33,45)(34,48)(35,47)(36,46)(49,52)(50,51)(57,60)(58,59) );
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)], [(1,15,10,50),(2,16,11,51),(3,13,12,52),(4,14,9,49),(5,63,37,36),(6,64,38,33),(7,61,39,34),(8,62,40,35),(17,44,46,22),(18,41,47,23),(19,42,48,24),(20,43,45,21),(25,57,54,30),(26,58,55,31),(27,59,56,32),(28,60,53,29)], [(1,46,26,35),(2,18,27,63),(3,48,28,33),(4,20,25,61),(5,16,23,59),(6,52,24,29),(7,14,21,57),(8,50,22,31),(9,45,54,34),(10,17,55,62),(11,47,56,36),(12,19,53,64),(13,42,60,38),(15,44,58,40),(30,39,49,43),(32,37,51,41)], [(1,11),(2,10),(3,9),(4,12),(5,44),(6,43),(7,42),(8,41),(13,14),(15,16),(17,63),(18,62),(19,61),(20,64),(21,38),(22,37),(23,40),(24,39),(25,53),(26,56),(27,55),(28,54),(29,30),(31,32),(33,45),(34,48),(35,47),(36,46),(49,52),(50,51),(57,60),(58,59)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | 2+ 1+4 |
| kernel | C42⋊20D4 | C4×C4⋊C4 | C24.3C22 | C23⋊2D4 | C2×C4×D4 | C2×C4⋊D4 | C2×C4⋊1D4 | C42 | C4⋊C4 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 3 | 4 | 8 | 4 | 4 | 2 |
Matrix representation of C42⋊20D4 ►in GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,3,0,0,0,0,1,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,4,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,3],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,1,1] >;
C42⋊20D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{20}D_4 % in TeX
G:=Group("C4^2:20D4"); // GroupNames label
G:=SmallGroup(128,1273);
// by ID
G=gap.SmallGroup(128,1273);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations