p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.73D4, C42.154C23, C4⋊Q8.18C4, C4⋊1D4.12C4, C42.95(C2×C4), (C22×C4).230D4, C42.6C4⋊39C2, C8⋊C4.89C22, C4.16(C4.D4), C23.60(C22⋊C4), (C2×C42).198C22, C42.C22⋊13C2, C4.4D4.119C22, C2.36(C42⋊C22), C22.26C24.11C2, (C2×C4○D4).5C4, (C2×D4).26(C2×C4), (C2×Q8).26(C2×C4), (C2×C4).1182(C2×D4), C2.14(C2×C4.D4), (C2×C4).99(C22⋊C4), (C2×C4).148(C22×C4), (C22×C4).220(C2×C4), C22.212(C2×C22⋊C4), SmallGroup(128,268)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.73D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 276 in 119 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C42.C22, C42.6C4, C22.26C24, C42.73D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C2×C4.D4, C42⋊C22, C42.73D4
Character table of C42.73D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | -i | i | -i | i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | i | -i | i | i | -i | -i | -i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | i | -i | i | -i | -i | i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | -i | i | -i | -i | i | i | i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | -i | i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | -i | i | -i | i | i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | i | -i | i | -i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | i | -i | i | -i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4i | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
►On 64 points
Generators in S64
(1 35 56 48)(2 41 49 36)(3 37 50 42)(4 43 51 38)(5 39 52 44)(6 45 53 40)(7 33 54 46)(8 47 55 34)(9 32 20 58)(10 59 21 25)(11 26 22 60)(12 61 23 27)(13 28 24 62)(14 63 17 29)(15 30 18 64)(16 57 19 31)
(1 27 52 57)(2 32 53 62)(3 29 54 59)(4 26 55 64)(5 31 56 61)(6 28 49 58)(7 25 50 63)(8 30 51 60)(9 45 24 36)(10 42 17 33)(11 47 18 38)(12 44 19 35)(13 41 20 40)(14 46 21 37)(15 43 22 34)(16 48 23 39)
(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 30 27 51 52 60 57 8)(2 54 32 59 53 3 62 29)(4 5 26 31 55 56 64 61)(6 50 28 63 49 7 58 25)(9 21 45 37 24 14 36 46)(10 40 42 13 17 41 33 20)(11 16 47 48 18 23 38 39)(12 43 44 22 19 34 35 15)
G:=sub<Sym(64)| (1,35,56,48)(2,41,49,36)(3,37,50,42)(4,43,51,38)(5,39,52,44)(6,45,53,40)(7,33,54,46)(8,47,55,34)(9,32,20,58)(10,59,21,25)(11,26,22,60)(12,61,23,27)(13,28,24,62)(14,63,17,29)(15,30,18,64)(16,57,19,31), (1,27,52,57)(2,32,53,62)(3,29,54,59)(4,26,55,64)(5,31,56,61)(6,28,49,58)(7,25,50,63)(8,30,51,60)(9,45,24,36)(10,42,17,33)(11,47,18,38)(12,44,19,35)(13,41,20,40)(14,46,21,37)(15,43,22,34)(16,48,23,39), (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), (1,30,27,51,52,60,57,8)(2,54,32,59,53,3,62,29)(4,5,26,31,55,56,64,61)(6,50,28,63,49,7,58,25)(9,21,45,37,24,14,36,46)(10,40,42,13,17,41,33,20)(11,16,47,48,18,23,38,39)(12,43,44,22,19,34,35,15)>;
G:=Group( (1,35,56,48)(2,41,49,36)(3,37,50,42)(4,43,51,38)(5,39,52,44)(6,45,53,40)(7,33,54,46)(8,47,55,34)(9,32,20,58)(10,59,21,25)(11,26,22,60)(12,61,23,27)(13,28,24,62)(14,63,17,29)(15,30,18,64)(16,57,19,31), (1,27,52,57)(2,32,53,62)(3,29,54,59)(4,26,55,64)(5,31,56,61)(6,28,49,58)(7,25,50,63)(8,30,51,60)(9,45,24,36)(10,42,17,33)(11,47,18,38)(12,44,19,35)(13,41,20,40)(14,46,21,37)(15,43,22,34)(16,48,23,39), (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), (1,30,27,51,52,60,57,8)(2,54,32,59,53,3,62,29)(4,5,26,31,55,56,64,61)(6,50,28,63,49,7,58,25)(9,21,45,37,24,14,36,46)(10,40,42,13,17,41,33,20)(11,16,47,48,18,23,38,39)(12,43,44,22,19,34,35,15) );
G=PermutationGroup([[(1,35,56,48),(2,41,49,36),(3,37,50,42),(4,43,51,38),(5,39,52,44),(6,45,53,40),(7,33,54,46),(8,47,55,34),(9,32,20,58),(10,59,21,25),(11,26,22,60),(12,61,23,27),(13,28,24,62),(14,63,17,29),(15,30,18,64),(16,57,19,31)], [(1,27,52,57),(2,32,53,62),(3,29,54,59),(4,26,55,64),(5,31,56,61),(6,28,49,58),(7,25,50,63),(8,30,51,60),(9,45,24,36),(10,42,17,33),(11,47,18,38),(12,44,19,35),(13,41,20,40),(14,46,21,37),(15,43,22,34),(16,48,23,39)], [(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)], [(1,30,27,51,52,60,57,8),(2,54,32,59,53,3,62,29),(4,5,26,31,55,56,64,61),(6,50,28,63,49,7,58,25),(9,21,45,37,24,14,36,46),(10,40,42,13,17,41,33,20),(11,16,47,48,18,23,38,39),(12,43,44,22,19,34,35,15)]])
Matrix representation of C42.73D4 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 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 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 13 |
| 13 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 | 8 | 9 |
| 0 | 0 | 0 | 0 | 11 | 11 | 8 | 8 |
| 0 | 0 | 0 | 0 | 8 | 9 | 6 | 11 |
| 0 | 0 | 0 | 0 | 8 | 8 | 6 | 6 |
| 13 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 | 8 | 9 |
| 0 | 0 | 0 | 0 | 6 | 6 | 9 | 9 |
| 0 | 0 | 0 | 0 | 8 | 9 | 6 | 11 |
| 0 | 0 | 0 | 0 | 9 | 9 | 11 | 11 |
G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[13,4,0,0,0,0,0,0,12,4,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,5,13,0,0,0,0,0,0,0,0,11,11,8,8,0,0,0,0,6,11,9,8,0,0,0,0,8,8,6,6,0,0,0,0,9,8,11,6],[13,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,16,0,0,0,0,0,0,0,0,11,6,8,9,0,0,0,0,6,6,9,9,0,0,0,0,8,9,6,11,0,0,0,0,9,9,11,11] >;
C42.73D4 in GAP, Magma, Sage, TeX
C_4^2._{73}D_4 % in TeX
G:=Group("C4^2.73D4"); // GroupNames label
G:=SmallGroup(128,268);
// by ID
G=gap.SmallGroup(128,268);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations
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