p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊21D4, C23.444C24, C24.324C23, C22.2322+ 1+4, (C2×D4).212D4, C23.50(C2×D4), C4.58(C4⋊D4), C2.70(D4⋊5D4), C23.23D4⋊55C2, C23.10D4⋊41C2, (C2×C42).549C22, (C22×C4).536C23, (C23×C4).113C22, C22.295(C22×D4), C24.3C22⋊55C2, (C22×D4).165C22, (C22×Q8).130C22, C23.67C23⋊58C2, C2.22(C22.29C24), C2.C42.182C22, C2.31(C22.26C24), C2.16(C22.53C24), (C2×C4×D4)⋊47C2, (C4×C4⋊C4)⋊88C2, (C2×C4).353(C2×D4), C2.36(C2×C4⋊D4), (C2×C4.4D4)⋊15C2, (C2×C4⋊1D4).16C2, (C2×C4).147(C4○D4), (C2×C4⋊C4).868C22, C22.321(C2×C4○D4), (C2×C22⋊C4).178C22, SmallGroup(128,1276)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊21D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 756 in 352 conjugacy classes, 112 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, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4.4D4, C4⋊1D4, C23×C4, C22×D4, C22×D4, C22×Q8, C4×C4⋊C4, C23.23D4, C24.3C22, C23.67C23, C23.10D4, C2×C4×D4, C2×C4.4D4, C2×C4⋊1D4, C42⋊21D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4⋊D4, C22.26C24, C22.29C24, D4⋊5D4, C22.53C24, C42⋊21D4
►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 12 21 43)(2 9 22 44)(3 10 23 41)(4 11 24 42)(5 56 40 26)(6 53 37 27)(7 54 38 28)(8 55 39 25)(13 64 48 36)(14 61 45 33)(15 62 46 34)(16 63 47 35)(17 30 52 60)(18 31 49 57)(19 32 50 58)(20 29 51 59)
(1 47 39 59)(2 46 40 58)(3 45 37 57)(4 48 38 60)(5 32 22 15)(6 31 23 14)(7 30 24 13)(8 29 21 16)(9 34 26 19)(10 33 27 18)(11 36 28 17)(12 35 25 20)(41 61 53 49)(42 64 54 52)(43 63 55 51)(44 62 56 50)
(1 39)(2 7)(3 37)(4 5)(6 23)(8 21)(9 28)(10 53)(11 26)(12 55)(13 46)(15 48)(17 19)(18 49)(20 51)(22 38)(24 40)(25 43)(27 41)(30 58)(32 60)(33 61)(34 36)(35 63)(42 56)(44 54)(50 52)(62 64)
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,36)(14,61,45,33)(15,62,46,34)(16,63,47,35)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59), (1,47,39,59)(2,46,40,58)(3,45,37,57)(4,48,38,60)(5,32,22,15)(6,31,23,14)(7,30,24,13)(8,29,21,16)(9,34,26,19)(10,33,27,18)(11,36,28,17)(12,35,25,20)(41,61,53,49)(42,64,54,52)(43,63,55,51)(44,62,56,50), (1,39)(2,7)(3,37)(4,5)(6,23)(8,21)(9,28)(10,53)(11,26)(12,55)(13,46)(15,48)(17,19)(18,49)(20,51)(22,38)(24,40)(25,43)(27,41)(30,58)(32,60)(33,61)(34,36)(35,63)(42,56)(44,54)(50,52)(62,64)>;
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,36)(14,61,45,33)(15,62,46,34)(16,63,47,35)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59), (1,47,39,59)(2,46,40,58)(3,45,37,57)(4,48,38,60)(5,32,22,15)(6,31,23,14)(7,30,24,13)(8,29,21,16)(9,34,26,19)(10,33,27,18)(11,36,28,17)(12,35,25,20)(41,61,53,49)(42,64,54,52)(43,63,55,51)(44,62,56,50), (1,39)(2,7)(3,37)(4,5)(6,23)(8,21)(9,28)(10,53)(11,26)(12,55)(13,46)(15,48)(17,19)(18,49)(20,51)(22,38)(24,40)(25,43)(27,41)(30,58)(32,60)(33,61)(34,36)(35,63)(42,56)(44,54)(50,52)(62,64) );
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)], [(1,12,21,43),(2,9,22,44),(3,10,23,41),(4,11,24,42),(5,56,40,26),(6,53,37,27),(7,54,38,28),(8,55,39,25),(13,64,48,36),(14,61,45,33),(15,62,46,34),(16,63,47,35),(17,30,52,60),(18,31,49,57),(19,32,50,58),(20,29,51,59)], [(1,47,39,59),(2,46,40,58),(3,45,37,57),(4,48,38,60),(5,32,22,15),(6,31,23,14),(7,30,24,13),(8,29,21,16),(9,34,26,19),(10,33,27,18),(11,36,28,17),(12,35,25,20),(41,61,53,49),(42,64,54,52),(43,63,55,51),(44,62,56,50)], [(1,39),(2,7),(3,37),(4,5),(6,23),(8,21),(9,28),(10,53),(11,26),(12,55),(13,46),(15,48),(17,19),(18,49),(20,51),(22,38),(24,40),(25,43),(27,41),(30,58),(32,60),(33,61),(34,36),(35,63),(42,56),(44,54),(50,52),(62,64)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2+ 1+4 |
| kernel | C42⋊21D4 | C4×C4⋊C4 | C23.23D4 | C24.3C22 | C23.67C23 | C23.10D4 | C2×C4×D4 | C2×C4.4D4 | C2×C4⋊1D4 | C42 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42⋊21D4 ►in GL6(𝔽5)
| 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 | 4 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 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 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 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 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,3,0,0,0,0,4,3],[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],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4] >;
C42⋊21D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{21}D_4 % in TeX
G:=Group("C4^2:21D4"); // GroupNames label
G:=SmallGroup(128,1276);
// by ID
G=gap.SmallGroup(128,1276);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,184,675,80]);
// 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^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations