p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.118D4, C4⋊C4⋊20D4, C4.96(C4×D4), C4⋊1D4⋊17C4, (C2×C4).135D8, C4⋊1(D4⋊C4), C2.5(C4⋊D8), C22.48(C2×D8), C4.24(C4⋊1D4), C42.160(C2×C4), C2.5(C4⋊SD16), (C2×C4).120SD16, C23.808(C2×D4), (C22×C4).763D4, C4.67(C4.4D4), (C22×C8).57C22, C22.74(C2×SD16), C22.98(C8⋊C22), (C2×C42).328C22, (C22×D4).54C22, C22.153(C4⋊D4), (C22×C4).1418C23, C2.23(C23.37D4), C2.34(C24.3C22), (C4×C4⋊C4)⋊9C2, (C2×C4⋊C8)⋊17C2, (C2×D4⋊C4)⋊9C2, (C2×C4⋊1D4).8C2, (C2×D4).119(C2×C4), C2.23(C2×D4⋊C4), (C2×C4).1360(C2×D4), (C2×C4).870(C4○D4), (C2×C4⋊C4).778C22, (C2×C4).432(C22×C4), (C2×C4).260(C22⋊C4), C22.292(C2×C22⋊C4), SmallGroup(128,714)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.118D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=b-1, dcd=b-1c-1 >
Subgroups: 556 in 214 conjugacy classes, 72 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C4⋊1D4, C4⋊1D4, C22×C8, C22×D4, C22×D4, C4×C4⋊C4, C2×D4⋊C4, C2×C4⋊C8, C2×C4⋊1D4, C42.118D4
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, C8⋊C22, C24.3C22, C2×D4⋊C4, C23.37D4, C4⋊D8, C4⋊SD16, C42.118D4
►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 35 29 9)(2 36 30 10)(3 33 31 11)(4 34 32 12)(5 23 60 48)(6 24 57 45)(7 21 58 46)(8 22 59 47)(13 25 39 49)(14 26 40 50)(15 27 37 51)(16 28 38 52)(17 54 42 62)(18 55 43 63)(19 56 44 64)(20 53 41 61)
(1 24 39 53)(2 21 40 54)(3 22 37 55)(4 23 38 56)(5 52 19 34)(6 49 20 35)(7 50 17 36)(8 51 18 33)(9 57 25 41)(10 58 26 42)(11 59 27 43)(12 60 28 44)(13 61 29 45)(14 62 30 46)(15 63 31 47)(16 64 32 48)
(1 52)(2 51)(3 50)(4 49)(5 57)(6 60)(7 59)(8 58)(9 16)(10 15)(11 14)(12 13)(17 43)(18 42)(19 41)(20 44)(21 22)(23 24)(25 32)(26 31)(27 30)(28 29)(33 40)(34 39)(35 38)(36 37)(45 48)(46 47)(53 56)(54 55)(61 64)(62 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), (1,35,29,9)(2,36,30,10)(3,33,31,11)(4,34,32,12)(5,23,60,48)(6,24,57,45)(7,21,58,46)(8,22,59,47)(13,25,39,49)(14,26,40,50)(15,27,37,51)(16,28,38,52)(17,54,42,62)(18,55,43,63)(19,56,44,64)(20,53,41,61), (1,24,39,53)(2,21,40,54)(3,22,37,55)(4,23,38,56)(5,52,19,34)(6,49,20,35)(7,50,17,36)(8,51,18,33)(9,57,25,41)(10,58,26,42)(11,59,27,43)(12,60,28,44)(13,61,29,45)(14,62,30,46)(15,63,31,47)(16,64,32,48), (1,52)(2,51)(3,50)(4,49)(5,57)(6,60)(7,59)(8,58)(9,16)(10,15)(11,14)(12,13)(17,43)(18,42)(19,41)(20,44)(21,22)(23,24)(25,32)(26,31)(27,30)(28,29)(33,40)(34,39)(35,38)(36,37)(45,48)(46,47)(53,56)(54,55)(61,64)(62,63)>;
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), (1,35,29,9)(2,36,30,10)(3,33,31,11)(4,34,32,12)(5,23,60,48)(6,24,57,45)(7,21,58,46)(8,22,59,47)(13,25,39,49)(14,26,40,50)(15,27,37,51)(16,28,38,52)(17,54,42,62)(18,55,43,63)(19,56,44,64)(20,53,41,61), (1,24,39,53)(2,21,40,54)(3,22,37,55)(4,23,38,56)(5,52,19,34)(6,49,20,35)(7,50,17,36)(8,51,18,33)(9,57,25,41)(10,58,26,42)(11,59,27,43)(12,60,28,44)(13,61,29,45)(14,62,30,46)(15,63,31,47)(16,64,32,48), (1,52)(2,51)(3,50)(4,49)(5,57)(6,60)(7,59)(8,58)(9,16)(10,15)(11,14)(12,13)(17,43)(18,42)(19,41)(20,44)(21,22)(23,24)(25,32)(26,31)(27,30)(28,29)(33,40)(34,39)(35,38)(36,37)(45,48)(46,47)(53,56)(54,55)(61,64)(62,63) );
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)], [(1,35,29,9),(2,36,30,10),(3,33,31,11),(4,34,32,12),(5,23,60,48),(6,24,57,45),(7,21,58,46),(8,22,59,47),(13,25,39,49),(14,26,40,50),(15,27,37,51),(16,28,38,52),(17,54,42,62),(18,55,43,63),(19,56,44,64),(20,53,41,61)], [(1,24,39,53),(2,21,40,54),(3,22,37,55),(4,23,38,56),(5,52,19,34),(6,49,20,35),(7,50,17,36),(8,51,18,33),(9,57,25,41),(10,58,26,42),(11,59,27,43),(12,60,28,44),(13,61,29,45),(14,62,30,46),(15,63,31,47),(16,64,32,48)], [(1,52),(2,51),(3,50),(4,49),(5,57),(6,60),(7,59),(8,58),(9,16),(10,15),(11,14),(12,13),(17,43),(18,42),(19,41),(20,44),(21,22),(23,24),(25,32),(26,31),(27,30),(28,29),(33,40),(34,39),(35,38),(36,37),(45,48),(46,47),(53,56),(54,55),(61,64),(62,63)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D8 | SD16 | C4○D4 | C8⋊C22 |
| kernel | C42.118D4 | C4×C4⋊C4 | C2×D4⋊C4 | C2×C4⋊C8 | C2×C4⋊1D4 | C4⋊1D4 | C42 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 2 | 4 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C42.118D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 13 |
| 0 | 0 | 0 | 0 | 11 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 13 |
| 0 | 0 | 0 | 0 | 2 | 3 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,11,0,0,0,0,13,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,14,2,0,0,0,0,13,3] >;
C42.118D4 in GAP, Magma, Sage, TeX
C_4^2._{118}D_4 % in TeX
G:=Group("C4^2.118D4"); // GroupNames label
G:=SmallGroup(128,714);
// by ID
G=gap.SmallGroup(128,714);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,100,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^-1*c^-1>;
// generators/relations