p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊27D4, C23.519C24, C24.362C23, C22.2972+ 1+4, (C22×C4)⋊35D4, C42⋊8C4⋊50C2, C4.99(C4⋊D4), C23⋊2D4⋊25C2, C23.191(C2×D4), C23.10D4⋊56C2, (C2×C42).600C22, (C23×C4).422C22, (C22×C4).129C23, C22.344(C22×D4), C24.3C22⋊64C2, (C22×D4).191C22, C2.34(C22.29C24), C2.C42.246C22, C2.24(C22.34C24), (C2×C4⋊1D4)⋊7C2, (C2×C4⋊D4)⋊23C2, (C2×C4).379(C2×D4), C2.43(C2×C4⋊D4), (C2×C42⋊C2)⋊36C2, (C2×C4).655(C4○D4), (C2×C4⋊C4).886C22, C22.391(C2×C4○D4), (C2×C22⋊C4).211C22, SmallGroup(128,1351)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊27D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 852 in 364 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C4⋊1D4, C23×C4, C22×D4, C42⋊8C4, C24.3C22, C23⋊2D4, C23.10D4, C2×C42⋊C2, C2×C4⋊D4, C2×C4⋊1D4, C42⋊27D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4⋊D4, C22.29C24, C22.34C24, C42⋊27D4
►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 43 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 45 17 13)(6 46 18 14)(7 47 19 15)(8 48 20 16)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 36)(30 62 58 33)(31 63 59 34)(32 64 60 35)
(1 63 51 19)(2 33 52 6)(3 61 49 17)(4 35 50 8)(5 37 36 21)(7 39 34 23)(9 57 53 13)(10 32 54 48)(11 59 55 15)(12 30 56 46)(14 44 58 28)(16 42 60 26)(18 40 62 24)(20 38 64 22)(25 45 41 29)(27 47 43 31)
(1 15)(2 48)(3 13)(4 46)(5 41)(6 10)(7 43)(8 12)(9 17)(11 19)(14 38)(16 40)(18 42)(20 44)(21 29)(22 58)(23 31)(24 60)(25 36)(26 62)(27 34)(28 64)(30 50)(32 52)(33 54)(35 56)(37 45)(39 47)(49 57)(51 59)(53 61)(55 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), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,45,17,13)(6,46,18,14)(7,47,19,15)(8,48,20,16)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,36)(30,62,58,33)(31,63,59,34)(32,64,60,35), (1,63,51,19)(2,33,52,6)(3,61,49,17)(4,35,50,8)(5,37,36,21)(7,39,34,23)(9,57,53,13)(10,32,54,48)(11,59,55,15)(12,30,56,46)(14,44,58,28)(16,42,60,26)(18,40,62,24)(20,38,64,22)(25,45,41,29)(27,47,43,31), (1,15)(2,48)(3,13)(4,46)(5,41)(6,10)(7,43)(8,12)(9,17)(11,19)(14,38)(16,40)(18,42)(20,44)(21,29)(22,58)(23,31)(24,60)(25,36)(26,62)(27,34)(28,64)(30,50)(32,52)(33,54)(35,56)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)>;
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), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,45,17,13)(6,46,18,14)(7,47,19,15)(8,48,20,16)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,36)(30,62,58,33)(31,63,59,34)(32,64,60,35), (1,63,51,19)(2,33,52,6)(3,61,49,17)(4,35,50,8)(5,37,36,21)(7,39,34,23)(9,57,53,13)(10,32,54,48)(11,59,55,15)(12,30,56,46)(14,44,58,28)(16,42,60,26)(18,40,62,24)(20,38,64,22)(25,45,41,29)(27,47,43,31), (1,15)(2,48)(3,13)(4,46)(5,41)(6,10)(7,43)(8,12)(9,17)(11,19)(14,38)(16,40)(18,42)(20,44)(21,29)(22,58)(23,31)(24,60)(25,36)(26,62)(27,34)(28,64)(30,50)(32,52)(33,54)(35,56)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63) );
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)], [(1,43,39,11),(2,44,40,12),(3,41,37,9),(4,42,38,10),(5,45,17,13),(6,46,18,14),(7,47,19,15),(8,48,20,16),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,36),(30,62,58,33),(31,63,59,34),(32,64,60,35)], [(1,63,51,19),(2,33,52,6),(3,61,49,17),(4,35,50,8),(5,37,36,21),(7,39,34,23),(9,57,53,13),(10,32,54,48),(11,59,55,15),(12,30,56,46),(14,44,58,28),(16,42,60,26),(18,40,62,24),(20,38,64,22),(25,45,41,29),(27,47,43,31)], [(1,15),(2,48),(3,13),(4,46),(5,41),(6,10),(7,43),(8,12),(9,17),(11,19),(14,38),(16,40),(18,42),(20,44),(21,29),(22,58),(23,31),(24,60),(25,36),(26,62),(27,34),(28,64),(30,50),(32,52),(33,54),(35,56),(37,45),(39,47),(49,57),(51,59),(53,61),(55,63)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2+ 1+4 |
| kernel | C42⋊27D4 | C42⋊8C4 | C24.3C22 | C23⋊2D4 | C23.10D4 | C2×C42⋊C2 | C2×C4⋊D4 | C2×C4⋊1D4 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 | 4 |
Matrix representation of C42⋊27D4 ►in GL8(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 |
| 0 | 0 | 0 | 0 | -2 | 0 | 2 | 1 |
| 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | -1 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | -1 |
| 1 | -2 | 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 | 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 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | -1 |
G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,-2,0,0,0,0,1,0,1,0,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,-1,1],[1,1,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,2,0,0,0,0,1,0,-1,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,2,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
C42⋊27D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{27}D_4 % in TeX
G:=Group("C4^2:27D4"); // GroupNames label
G:=SmallGroup(128,1351);
// by ID
G=gap.SmallGroup(128,1351);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,185,80]);
// 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^-1*b^2,d*a*d=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations