p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.432D4, (C2×C8)⋊29D4, C4.10(C4×D4), (C2×C4).75D8, C4⋊1D4⋊12C4, C4⋊2(D4⋊C4), C42⋊9C4⋊3C2, C2.2(C8⋊4D4), C2.3(C8⋊5D4), (C2×C4).73SD16, C22.46(C2×D8), C4.73(C4⋊D4), C42.266(C2×C4), C2.3(C4.4D8), C23.796(C2×D4), (C22×C4).580D4, C22.71(C2×SD16), C22.32(C4⋊1D4), (C22×C8).490C22, (C22×D4).41C22, (C22×C4).1401C23, (C2×C42).1072C22, C22.63(C4.4D4), C2.9(C24.3C22), (C2×C4×C8)⋊13C2, (C2×D4⋊C4)⋊7C2, (C2×C4).735(C2×D4), (C2×C4⋊1D4).5C2, (C2×D4).103(C2×C4), C2.22(C2×D4⋊C4), (C2×C4⋊C4).84C22, (C2×C4).592(C4○D4), (C2×C4).415(C22×C4), (C2×C4).256(C22⋊C4), C22.279(C2×C22⋊C4), SmallGroup(128,689)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.432D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=bc-1 >
Subgroups: 564 in 218 conjugacy classes, 80 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, D4⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊1D4, C4⋊1D4, C22×C8, C22×D4, C22×D4, C42⋊9C4, C2×C4×C8, C2×D4⋊C4, C2×C4⋊1D4, C42.432D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4○D4, D4⋊C4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C24.3C22, C2×D4⋊C4, C4.4D8, C8⋊5D4, C8⋊4D4, C42.432D4
►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 39 29 13)(2 40 30 14)(3 37 31 15)(4 38 32 16)(5 19 60 44)(6 20 57 41)(7 17 58 42)(8 18 59 43)(9 49 35 25)(10 50 36 26)(11 51 33 27)(12 52 34 28)(21 62 46 54)(22 63 47 55)(23 64 48 56)(24 61 45 53)
(1 56 9 44)(2 55 10 43)(3 54 11 42)(4 53 12 41)(5 13 23 25)(6 16 24 28)(7 15 21 27)(8 14 22 26)(17 31 62 33)(18 30 63 36)(19 29 64 35)(20 32 61 34)(37 46 51 58)(38 45 52 57)(39 48 49 60)(40 47 50 59)
(1 4)(2 3)(5 61)(6 64)(7 63)(8 62)(9 12)(10 11)(13 38)(14 37)(15 40)(16 39)(17 22)(18 21)(19 24)(20 23)(25 52)(26 51)(27 50)(28 49)(29 32)(30 31)(33 36)(34 35)(41 48)(42 47)(43 46)(44 45)(53 60)(54 59)(55 58)(56 57)
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), (1,39,29,13)(2,40,30,14)(3,37,31,15)(4,38,32,16)(5,19,60,44)(6,20,57,41)(7,17,58,42)(8,18,59,43)(9,49,35,25)(10,50,36,26)(11,51,33,27)(12,52,34,28)(21,62,46,54)(22,63,47,55)(23,64,48,56)(24,61,45,53), (1,56,9,44)(2,55,10,43)(3,54,11,42)(4,53,12,41)(5,13,23,25)(6,16,24,28)(7,15,21,27)(8,14,22,26)(17,31,62,33)(18,30,63,36)(19,29,64,35)(20,32,61,34)(37,46,51,58)(38,45,52,57)(39,48,49,60)(40,47,50,59), (1,4)(2,3)(5,61)(6,64)(7,63)(8,62)(9,12)(10,11)(13,38)(14,37)(15,40)(16,39)(17,22)(18,21)(19,24)(20,23)(25,52)(26,51)(27,50)(28,49)(29,32)(30,31)(33,36)(34,35)(41,48)(42,47)(43,46)(44,45)(53,60)(54,59)(55,58)(56,57)>;
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), (1,39,29,13)(2,40,30,14)(3,37,31,15)(4,38,32,16)(5,19,60,44)(6,20,57,41)(7,17,58,42)(8,18,59,43)(9,49,35,25)(10,50,36,26)(11,51,33,27)(12,52,34,28)(21,62,46,54)(22,63,47,55)(23,64,48,56)(24,61,45,53), (1,56,9,44)(2,55,10,43)(3,54,11,42)(4,53,12,41)(5,13,23,25)(6,16,24,28)(7,15,21,27)(8,14,22,26)(17,31,62,33)(18,30,63,36)(19,29,64,35)(20,32,61,34)(37,46,51,58)(38,45,52,57)(39,48,49,60)(40,47,50,59), (1,4)(2,3)(5,61)(6,64)(7,63)(8,62)(9,12)(10,11)(13,38)(14,37)(15,40)(16,39)(17,22)(18,21)(19,24)(20,23)(25,52)(26,51)(27,50)(28,49)(29,32)(30,31)(33,36)(34,35)(41,48)(42,47)(43,46)(44,45)(53,60)(54,59)(55,58)(56,57) );
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)], [(1,39,29,13),(2,40,30,14),(3,37,31,15),(4,38,32,16),(5,19,60,44),(6,20,57,41),(7,17,58,42),(8,18,59,43),(9,49,35,25),(10,50,36,26),(11,51,33,27),(12,52,34,28),(21,62,46,54),(22,63,47,55),(23,64,48,56),(24,61,45,53)], [(1,56,9,44),(2,55,10,43),(3,54,11,42),(4,53,12,41),(5,13,23,25),(6,16,24,28),(7,15,21,27),(8,14,22,26),(17,31,62,33),(18,30,63,36),(19,29,64,35),(20,32,61,34),(37,46,51,58),(38,45,52,57),(39,48,49,60),(40,47,50,59)], [(1,4),(2,3),(5,61),(6,64),(7,63),(8,62),(9,12),(10,11),(13,38),(14,37),(15,40),(16,39),(17,22),(18,21),(19,24),(20,23),(25,52),(26,51),(27,50),(28,49),(29,32),(30,31),(33,36),(34,35),(41,48),(42,47),(43,46),(44,45),(53,60),(54,59),(55,58),(56,57)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D8 | SD16 | C4○D4 |
| kernel | C42.432D4 | C42⋊9C4 | C2×C4×C8 | C2×D4⋊C4 | C2×C4⋊1D4 | C4⋊1D4 | C42 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 8 | 2 | 4 | 2 | 8 | 8 | 4 |
Matrix representation of C42.432D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 3 | 14 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,0],[4,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,3,3,0,0,0,3,14],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,16,0] >;
C42.432D4 in GAP, Magma, Sage, TeX
C_4^2._{432}D_4 % in TeX
G:=Group("C4^2.432D4"); // GroupNames label
G:=SmallGroup(128,689);
// by ID
G=gap.SmallGroup(128,689);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,100,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b*c^-1>;
// generators/relations