p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.160D4, C23.208C24, C24.199C23, C22.312- 1+4, C22.472+ 1+4, C4.37(C4×D4), C42⋊18(C2×C4), C4.4D4⋊22C4, C42⋊9C4⋊11C2, C42⋊4C4⋊11C2, C23.8Q8⋊9C2, C23.12(C22×C4), (C23×C4).47C22, C22.99(C23×C4), C22.96(C22×D4), C24.C22⋊8C2, (C22×C4).473C23, (C2×C42).415C22, C2.6(C22.29C24), (C22×D4).479C22, (C22×Q8).399C22, C23.67C23⋊14C2, C2.C42.44C22, C24.3C22.25C2, C2.6(C23.38C23), C2.6(C22.36C24), C2.15(C23.33C23), (C2×C4×Q8)⋊5C2, C2.25(C2×C4×D4), (C2×C4×D4).32C2, (C2×Q8)⋊24(C2×C4), C22⋊C4⋊12(C2×C4), (C2×D4).168(C2×C4), (C2×C4).1188(C2×D4), (C2×C4).29(C22×C4), C22.93(C2×C4○D4), (C2×C4).648(C4○D4), (C2×C4⋊C4).180C22, (C2×C4.4D4).15C2, (C2×C22⋊C4).28C22, SmallGroup(128,1058)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.160D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 540 in 306 conjugacy classes, 148 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, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C4.4D4, C23×C4, C22×D4, C22×Q8, C42⋊4C4, C42⋊9C4, C23.8Q8, C24.C22, C24.3C22, C23.67C23, C2×C4×D4, C2×C4×Q8, C2×C4.4D4, C42.160D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4×D4, C23.33C23, C22.29C24, C23.38C23, C22.36C24, C42.160D4
►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 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 18 38 46)(6 19 39 47)(7 20 40 48)(8 17 37 45)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(1 59 51 47)(2 32 52 20)(3 57 49 45)(4 30 50 18)(5 42 36 54)(6 15 33 27)(7 44 34 56)(8 13 35 25)(9 29 21 17)(10 58 22 46)(11 31 23 19)(12 60 24 48)(14 62 26 38)(16 64 28 40)(37 41 61 53)(39 43 63 55)
(1 26 3 28)(2 27 4 25)(5 17 7 19)(6 18 8 20)(9 56 11 54)(10 53 12 55)(13 52 15 50)(14 49 16 51)(21 44 23 42)(22 41 24 43)(29 34 31 36)(30 35 32 33)(37 48 39 46)(38 45 40 47)(57 64 59 62)(58 61 60 63)
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,59,51,47)(2,32,52,20)(3,57,49,45)(4,30,50,18)(5,42,36,54)(6,15,33,27)(7,44,34,56)(8,13,35,25)(9,29,21,17)(10,58,22,46)(11,31,23,19)(12,60,24,48)(14,62,26,38)(16,64,28,40)(37,41,61,53)(39,43,63,55), (1,26,3,28)(2,27,4,25)(5,17,7,19)(6,18,8,20)(9,56,11,54)(10,53,12,55)(13,52,15,50)(14,49,16,51)(21,44,23,42)(22,41,24,43)(29,34,31,36)(30,35,32,33)(37,48,39,46)(38,45,40,47)(57,64,59,62)(58,61,60,63)>;
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,59,51,47)(2,32,52,20)(3,57,49,45)(4,30,50,18)(5,42,36,54)(6,15,33,27)(7,44,34,56)(8,13,35,25)(9,29,21,17)(10,58,22,46)(11,31,23,19)(12,60,24,48)(14,62,26,38)(16,64,28,40)(37,41,61,53)(39,43,63,55), (1,26,3,28)(2,27,4,25)(5,17,7,19)(6,18,8,20)(9,56,11,54)(10,53,12,55)(13,52,15,50)(14,49,16,51)(21,44,23,42)(22,41,24,43)(29,34,31,36)(30,35,32,33)(37,48,39,46)(38,45,40,47)(57,64,59,62)(58,61,60,63) );
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)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,18,38,46),(6,19,39,47),(7,20,40,48),(8,17,37,45),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,35),(30,62,58,36),(31,63,59,33),(32,64,60,34)], [(1,59,51,47),(2,32,52,20),(3,57,49,45),(4,30,50,18),(5,42,36,54),(6,15,33,27),(7,44,34,56),(8,13,35,25),(9,29,21,17),(10,58,22,46),(11,31,23,19),(12,60,24,48),(14,62,26,38),(16,64,28,40),(37,41,61,53),(39,43,63,55)], [(1,26,3,28),(2,27,4,25),(5,17,7,19),(6,18,8,20),(9,56,11,54),(10,53,12,55),(13,52,15,50),(14,49,16,51),(21,44,23,42),(22,41,24,43),(29,34,31,36),(30,35,32,33),(37,48,39,46),(38,45,40,47),(57,64,59,62),(58,61,60,63)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.160D4 | C42⋊4C4 | C42⋊9C4 | C23.8Q8 | C24.C22 | C24.3C22 | C23.67C23 | C2×C4×D4 | C2×C4×Q8 | C2×C4.4D4 | C4.4D4 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 16 | 4 | 4 | 2 | 2 |
Matrix representation of C42.160D4 ►in GL8(𝔽5)
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 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 | 4 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 1 |
| 0 | 0 | 0 | 0 | 3 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 4 |
| 0 | 0 | 0 | 0 | 2 | 0 | 4 | 0 |
G:=sub<GL(8,GF(5))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,4,0,3,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,3,0,3],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,1,0,2,0,0,0,0,0,0,4,0,4,0,0,0,0,4,0,4,0] >;
C42.160D4 in GAP, Magma, Sage, TeX
C_4^2._{160}D_4 % in TeX
G:=Group("C4^2.160D4"); // GroupNames label
G:=SmallGroup(128,1058);
// by ID
G=gap.SmallGroup(128,1058);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations