p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4⋊7D4, (C2×C4).21D8, (C2×D4).105D4, (C2×C4).34SD16, C22.84(C2×D8), C2.13(C4⋊D8), C4.22(C4⋊D4), C2.6(C4.4D8), C23.913(C2×D4), (C22×C4).313D4, C2.22(C22⋊D8), C2.13(C4⋊SD16), (C22×C8).73C22, C22.97(C2×SD16), C22.223C22≀C2, C2.22(C22⋊SD16), (C2×C42).364C22, (C22×D4).79C22, C22.230(C4⋊D4), C22.138(C8⋊C22), C22.7C42⋊11C2, (C22×C4).1447C23, C23.65C23⋊5C2, C22.90(C4.4D4), C4.71(C22.D4), C2.14(C23.10D4), C2.6(C42.29C22), (C2×D4⋊C4)⋊14C2, (C2×C4⋊1D4).10C2, (C2×C4).1039(C2×D4), (C2×C4).878(C4○D4), (C2×C4⋊C4).126C22, SmallGroup(128,773)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4⋊7D4
G = < a,b,c,d | a4=b4=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b-1, dbd=ab, dcd=c-1 >
Subgroups: 552 in 201 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊1D4, C22×C8, C22×D4, C22×D4, C22.7C42, C23.65C23, C2×D4⋊C4, C2×C4⋊1D4, C4⋊C4⋊7D4
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×D8, C2×SD16, C8⋊C22, C23.10D4, C22⋊D8, C22⋊SD16, C4⋊D8, C4⋊SD16, C4.4D8, C42.29C22, C4⋊C4⋊7D4
►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 63 11 58)(2 62 12 57)(3 61 9 60)(4 64 10 59)(5 13 21 52)(6 16 22 51)(7 15 23 50)(8 14 24 49)(17 36 39 53)(18 35 40 56)(19 34 37 55)(20 33 38 54)(25 44 30 45)(26 43 31 48)(27 42 32 47)(28 41 29 46)
(1 34 8 44)(2 35 5 41)(3 36 6 42)(4 33 7 43)(9 53 22 47)(10 54 23 48)(11 55 24 45)(12 56 21 46)(13 28 62 18)(14 25 63 19)(15 26 64 20)(16 27 61 17)(29 57 40 52)(30 58 37 49)(31 59 38 50)(32 60 39 51)
(1 44)(2 43)(3 42)(4 41)(5 33)(6 36)(7 35)(8 34)(9 47)(10 46)(11 45)(12 48)(13 37)(14 40)(15 39)(16 38)(17 50)(18 49)(19 52)(20 51)(21 54)(22 53)(23 56)(24 55)(25 57)(26 60)(27 59)(28 58)(29 63)(30 62)(31 61)(32 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), (1,63,11,58)(2,62,12,57)(3,61,9,60)(4,64,10,59)(5,13,21,52)(6,16,22,51)(7,15,23,50)(8,14,24,49)(17,36,39,53)(18,35,40,56)(19,34,37,55)(20,33,38,54)(25,44,30,45)(26,43,31,48)(27,42,32,47)(28,41,29,46), (1,34,8,44)(2,35,5,41)(3,36,6,42)(4,33,7,43)(9,53,22,47)(10,54,23,48)(11,55,24,45)(12,56,21,46)(13,28,62,18)(14,25,63,19)(15,26,64,20)(16,27,61,17)(29,57,40,52)(30,58,37,49)(31,59,38,50)(32,60,39,51), (1,44)(2,43)(3,42)(4,41)(5,33)(6,36)(7,35)(8,34)(9,47)(10,46)(11,45)(12,48)(13,37)(14,40)(15,39)(16,38)(17,50)(18,49)(19,52)(20,51)(21,54)(22,53)(23,56)(24,55)(25,57)(26,60)(27,59)(28,58)(29,63)(30,62)(31,61)(32,64)>;
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), (1,63,11,58)(2,62,12,57)(3,61,9,60)(4,64,10,59)(5,13,21,52)(6,16,22,51)(7,15,23,50)(8,14,24,49)(17,36,39,53)(18,35,40,56)(19,34,37,55)(20,33,38,54)(25,44,30,45)(26,43,31,48)(27,42,32,47)(28,41,29,46), (1,34,8,44)(2,35,5,41)(3,36,6,42)(4,33,7,43)(9,53,22,47)(10,54,23,48)(11,55,24,45)(12,56,21,46)(13,28,62,18)(14,25,63,19)(15,26,64,20)(16,27,61,17)(29,57,40,52)(30,58,37,49)(31,59,38,50)(32,60,39,51), (1,44)(2,43)(3,42)(4,41)(5,33)(6,36)(7,35)(8,34)(9,47)(10,46)(11,45)(12,48)(13,37)(14,40)(15,39)(16,38)(17,50)(18,49)(19,52)(20,51)(21,54)(22,53)(23,56)(24,55)(25,57)(26,60)(27,59)(28,58)(29,63)(30,62)(31,61)(32,64) );
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)], [(1,63,11,58),(2,62,12,57),(3,61,9,60),(4,64,10,59),(5,13,21,52),(6,16,22,51),(7,15,23,50),(8,14,24,49),(17,36,39,53),(18,35,40,56),(19,34,37,55),(20,33,38,54),(25,44,30,45),(26,43,31,48),(27,42,32,47),(28,41,29,46)], [(1,34,8,44),(2,35,5,41),(3,36,6,42),(4,33,7,43),(9,53,22,47),(10,54,23,48),(11,55,24,45),(12,56,21,46),(13,28,62,18),(14,25,63,19),(15,26,64,20),(16,27,61,17),(29,57,40,52),(30,58,37,49),(31,59,38,50),(32,60,39,51)], [(1,44),(2,43),(3,42),(4,41),(5,33),(6,36),(7,35),(8,34),(9,47),(10,46),(11,45),(12,48),(13,37),(14,40),(15,39),(16,38),(17,50),(18,49),(19,52),(20,51),(21,54),(22,53),(23,56),(24,55),(25,57),(26,60),(27,59),(28,58),(29,63),(30,62),(31,61),(32,64)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | SD16 | C4○D4 | C8⋊C22 |
| kernel | C4⋊C4⋊7D4 | C22.7C42 | C23.65C23 | C2×D4⋊C4 | C2×C4⋊1D4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 4 | 4 | 4 | 6 | 2 |
Matrix representation of C4⋊C4⋊7D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 8 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,1,8,0,0,0,0,4,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,13,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,13,1] >;
C4⋊C4⋊7D4 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_7D_4
% in TeX
G:=Group("C4:C4:7D4"); // GroupNames label
G:=SmallGroup(128,773);
// by ID
G=gap.SmallGroup(128,773);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2804,1411,718,172,4037,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations