p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.110D4, C4.12(C4×D4), (C2×C8).204D4, C42⋊9C4⋊4C2, C4.4D4⋊15C4, C2.2(C8⋊3D4), C4.75(C4⋊D4), C42.154(C2×C4), C2.2(C8.2D4), C23.798(C2×D4), (C22×C4).296D4, C22.34(C4⋊1D4), C22.94(C8⋊C22), (C2×C42).315C22, (C22×C8).403C22, (C22×D4).43C22, (C22×Q8).34C22, (C22×C4).1403C23, C22.65(C4.4D4), C22.83(C8.C22), C2.28(C23.36D4), C2.11(C24.3C22), C2.4(C42.28C22), (C2×C8⋊C4)⋊27C2, (C2×C4).737(C2×D4), (C2×Q8).90(C2×C4), (C2×Q8⋊C4)⋊47C2, (C2×D4).105(C2×C4), (C2×C4⋊C4).86C22, (C2×C4.4D4).8C2, (C2×D4⋊C4).35C2, (C2×C4).594(C4○D4), (C2×C4).417(C22×C4), (C2×C4).136(C22⋊C4), C22.281(C2×C22⋊C4), SmallGroup(128,691)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.110D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >
Subgroups: 388 in 172 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4.4D4, C4.4D4, C22×C8, C22×D4, C22×Q8, C42⋊9C4, C2×C8⋊C4, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.4D4, C42.110D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C8⋊C22, C8.C22, C24.3C22, C23.36D4, C42.28C22, C8⋊3D4, C8.2D4, C42.110D4
►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 38 29 13)(2 39 30 14)(3 40 31 15)(4 37 32 16)(5 19 59 44)(6 20 60 41)(7 17 57 42)(8 18 58 43)(9 49 34 25)(10 50 35 26)(11 51 36 27)(12 52 33 28)(21 62 46 53)(22 63 47 54)(23 64 48 55)(24 61 45 56)
(1 55 9 44)(2 54 10 43)(3 53 11 42)(4 56 12 41)(5 13 23 25)(6 16 24 28)(7 15 21 27)(8 14 22 26)(17 31 62 36)(18 30 63 35)(19 29 64 34)(20 32 61 33)(37 45 52 60)(38 48 49 59)(39 47 50 58)(40 46 51 57)
(1 4 29 32)(2 31 30 3)(5 61 59 56)(6 55 60 64)(7 63 57 54)(8 53 58 62)(9 12 34 33)(10 36 35 11)(13 37 38 16)(14 15 39 40)(17 22 42 47)(18 46 43 21)(19 24 44 45)(20 48 41 23)(25 52 49 28)(26 27 50 51)
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), (1,38,29,13)(2,39,30,14)(3,40,31,15)(4,37,32,16)(5,19,59,44)(6,20,60,41)(7,17,57,42)(8,18,58,43)(9,49,34,25)(10,50,35,26)(11,51,36,27)(12,52,33,28)(21,62,46,53)(22,63,47,54)(23,64,48,55)(24,61,45,56), (1,55,9,44)(2,54,10,43)(3,53,11,42)(4,56,12,41)(5,13,23,25)(6,16,24,28)(7,15,21,27)(8,14,22,26)(17,31,62,36)(18,30,63,35)(19,29,64,34)(20,32,61,33)(37,45,52,60)(38,48,49,59)(39,47,50,58)(40,46,51,57), (1,4,29,32)(2,31,30,3)(5,61,59,56)(6,55,60,64)(7,63,57,54)(8,53,58,62)(9,12,34,33)(10,36,35,11)(13,37,38,16)(14,15,39,40)(17,22,42,47)(18,46,43,21)(19,24,44,45)(20,48,41,23)(25,52,49,28)(26,27,50,51)>;
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), (1,38,29,13)(2,39,30,14)(3,40,31,15)(4,37,32,16)(5,19,59,44)(6,20,60,41)(7,17,57,42)(8,18,58,43)(9,49,34,25)(10,50,35,26)(11,51,36,27)(12,52,33,28)(21,62,46,53)(22,63,47,54)(23,64,48,55)(24,61,45,56), (1,55,9,44)(2,54,10,43)(3,53,11,42)(4,56,12,41)(5,13,23,25)(6,16,24,28)(7,15,21,27)(8,14,22,26)(17,31,62,36)(18,30,63,35)(19,29,64,34)(20,32,61,33)(37,45,52,60)(38,48,49,59)(39,47,50,58)(40,46,51,57), (1,4,29,32)(2,31,30,3)(5,61,59,56)(6,55,60,64)(7,63,57,54)(8,53,58,62)(9,12,34,33)(10,36,35,11)(13,37,38,16)(14,15,39,40)(17,22,42,47)(18,46,43,21)(19,24,44,45)(20,48,41,23)(25,52,49,28)(26,27,50,51) );
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)], [(1,38,29,13),(2,39,30,14),(3,40,31,15),(4,37,32,16),(5,19,59,44),(6,20,60,41),(7,17,57,42),(8,18,58,43),(9,49,34,25),(10,50,35,26),(11,51,36,27),(12,52,33,28),(21,62,46,53),(22,63,47,54),(23,64,48,55),(24,61,45,56)], [(1,55,9,44),(2,54,10,43),(3,53,11,42),(4,56,12,41),(5,13,23,25),(6,16,24,28),(7,15,21,27),(8,14,22,26),(17,31,62,36),(18,30,63,35),(19,29,64,34),(20,32,61,33),(37,45,52,60),(38,48,49,59),(39,47,50,58),(40,46,51,57)], [(1,4,29,32),(2,31,30,3),(5,61,59,56),(6,55,60,64),(7,63,57,54),(8,53,58,62),(9,12,34,33),(10,36,35,11),(13,37,38,16),(14,15,39,40),(17,22,42,47),(18,46,43,21),(19,24,44,45),(20,48,41,23),(25,52,49,28),(26,27,50,51)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.110D4 | C42⋊9C4 | C2×C8⋊C4 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.4D4 | C4.4D4 | C42 | C2×C8 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 2 | 4 | 2 | 4 | 2 | 2 |
Matrix representation of C42.110D4 ►in GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 16 |
| 1 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[1,9,0,0,0,0,0,0,13,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0],[16,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4] >;
C42.110D4 in GAP, Magma, Sage, TeX
C_4^2._{110}D_4 % in TeX
G:=Group("C4^2.110D4"); // GroupNames label
G:=SmallGroup(128,691);
// by ID
G=gap.SmallGroup(128,691);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;
// generators/relations