p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊31D4, C24.52C23, C23.557C24, C22.3322+ 1+4, C4.7(C4⋊1D4), C42⋊4C4⋊33C2, C23⋊2D4⋊29C2, (C22×C4).856C23, (C2×C42).622C22, C22.369(C22×D4), (C22×D4).207C22, (C22×Q8).164C22, C2.47(C22.29C24), C2.C42.560C22, (C2×C4⋊1D4)⋊8C2, (C2×C4).403(C2×D4), C2.14(C2×C4⋊1D4), (C2×C4.4D4)⋊22C2, (C2×C22⋊C4).237C22, SmallGroup(128,1389)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊31D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1028 in 422 conjugacy classes, 116 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C4.4D4, C4⋊1D4, C22×D4, C22×Q8, C42⋊4C4, C23⋊2D4, C2×C4.4D4, C2×C4⋊1D4, C42⋊31D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C22×D4, 2+ 1+4, C2×C4⋊1D4, C22.29C24, C42⋊31D4
►On 64 points
Generators in S64
(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 35 60 26)(2 36 57 27)(3 33 58 28)(4 34 59 25)(5 55 31 63)(6 56 32 64)(7 53 29 61)(8 54 30 62)(9 44 49 18)(10 41 50 19)(11 42 51 20)(12 43 52 17)(13 22 45 40)(14 23 46 37)(15 24 47 38)(16 21 48 39)
(1 42 7 14)(2 17 8 47)(3 44 5 16)(4 19 6 45)(9 63 39 28)(10 56 40 34)(11 61 37 26)(12 54 38 36)(13 59 41 32)(15 57 43 30)(18 31 48 58)(20 29 46 60)(21 33 49 55)(22 25 50 64)(23 35 51 53)(24 27 52 62)
(1 62)(2 61)(3 64)(4 63)(5 25)(6 28)(7 27)(8 26)(9 19)(10 18)(11 17)(12 20)(13 21)(14 24)(15 23)(16 22)(29 36)(30 35)(31 34)(32 33)(37 47)(38 46)(39 45)(40 48)(41 49)(42 52)(43 51)(44 50)(53 57)(54 60)(55 59)(56 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), (1,35,60,26)(2,36,57,27)(3,33,58,28)(4,34,59,25)(5,55,31,63)(6,56,32,64)(7,53,29,61)(8,54,30,62)(9,44,49,18)(10,41,50,19)(11,42,51,20)(12,43,52,17)(13,22,45,40)(14,23,46,37)(15,24,47,38)(16,21,48,39), (1,42,7,14)(2,17,8,47)(3,44,5,16)(4,19,6,45)(9,63,39,28)(10,56,40,34)(11,61,37,26)(12,54,38,36)(13,59,41,32)(15,57,43,30)(18,31,48,58)(20,29,46,60)(21,33,49,55)(22,25,50,64)(23,35,51,53)(24,27,52,62), (1,62)(2,61)(3,64)(4,63)(5,25)(6,28)(7,27)(8,26)(9,19)(10,18)(11,17)(12,20)(13,21)(14,24)(15,23)(16,22)(29,36)(30,35)(31,34)(32,33)(37,47)(38,46)(39,45)(40,48)(41,49)(42,52)(43,51)(44,50)(53,57)(54,60)(55,59)(56,58)>;
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), (1,35,60,26)(2,36,57,27)(3,33,58,28)(4,34,59,25)(5,55,31,63)(6,56,32,64)(7,53,29,61)(8,54,30,62)(9,44,49,18)(10,41,50,19)(11,42,51,20)(12,43,52,17)(13,22,45,40)(14,23,46,37)(15,24,47,38)(16,21,48,39), (1,42,7,14)(2,17,8,47)(3,44,5,16)(4,19,6,45)(9,63,39,28)(10,56,40,34)(11,61,37,26)(12,54,38,36)(13,59,41,32)(15,57,43,30)(18,31,48,58)(20,29,46,60)(21,33,49,55)(22,25,50,64)(23,35,51,53)(24,27,52,62), (1,62)(2,61)(3,64)(4,63)(5,25)(6,28)(7,27)(8,26)(9,19)(10,18)(11,17)(12,20)(13,21)(14,24)(15,23)(16,22)(29,36)(30,35)(31,34)(32,33)(37,47)(38,46)(39,45)(40,48)(41,49)(42,52)(43,51)(44,50)(53,57)(54,60)(55,59)(56,58) );
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)], [(1,35,60,26),(2,36,57,27),(3,33,58,28),(4,34,59,25),(5,55,31,63),(6,56,32,64),(7,53,29,61),(8,54,30,62),(9,44,49,18),(10,41,50,19),(11,42,51,20),(12,43,52,17),(13,22,45,40),(14,23,46,37),(15,24,47,38),(16,21,48,39)], [(1,42,7,14),(2,17,8,47),(3,44,5,16),(4,19,6,45),(9,63,39,28),(10,56,40,34),(11,61,37,26),(12,54,38,36),(13,59,41,32),(15,57,43,30),(18,31,48,58),(20,29,46,60),(21,33,49,55),(22,25,50,64),(23,35,51,53),(24,27,52,62)], [(1,62),(2,61),(3,64),(4,63),(5,25),(6,28),(7,27),(8,26),(9,19),(10,18),(11,17),(12,20),(13,21),(14,24),(15,23),(16,22),(29,36),(30,35),(31,34),(32,33),(37,47),(38,46),(39,45),(40,48),(41,49),(42,52),(43,51),(44,50),(53,57),(54,60),(55,59),(56,58)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | D4 | 2+ 1+4 |
| kernel | C42⋊31D4 | C42⋊4C4 | C23⋊2D4 | C2×C4.4D4 | C2×C4⋊1D4 | C42 | C22 |
| # reps | 1 | 1 | 8 | 3 | 3 | 12 | 4 |
Matrix representation of C42⋊31D4 ►in GL8(𝔽5)
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
G:=sub<GL(8,GF(5))| [0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,2,0,0,0,0,3,2,0,0,0,0,0,0,0,2,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,3,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,1,2],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,1,2,0,0,0,0,3,2,0,0,0,0,0,0,1,2,0,0] >;
C42⋊31D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{31}D_4 % in TeX
G:=Group("C4^2:31D4"); // GroupNames label
G:=SmallGroup(128,1389);
// by ID
G=gap.SmallGroup(128,1389);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,185,80]);
// 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,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations