p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.234D4, C42.350C23, D4⋊2Q8⋊6C2, Q16⋊C4⋊7C2, C4.Q16⋊23C2, C4⋊2Q16⋊23C2, D4.8(C4○D4), D4.D4⋊6C2, C4⋊C8.52C22, C4⋊C4.69C23, (C2×C8).43C23, Q8.7(C4○D4), (C2×C4).314C24, C8⋊C4.9C22, C42.6C4⋊7C2, D4.7D4.1C2, (C22×C4).454D4, C23.254(C2×D4), C4⋊Q8.270C22, SD16⋊C4⋊10C2, (C4×Q8).75C22, (C2×Q8).79C23, C4.Q8.18C22, C2.D8.89C22, (C4×D4).322C22, (C2×D4).406C23, C22⋊C8.27C22, (C2×Q16).58C22, D4⋊C4.33C22, C23.20D4⋊18C2, C4.119(C8.C22), (C2×C42).841C22, Q8⋊C4.34C22, (C2×SD16).15C22, C22.574(C22×D4), C22⋊Q8.173C22, C2.33(D8⋊C22), (C22×C4).1030C23, C23.37C23⋊7C2, C42⋊C2.324C22, C2.115(C22.19C24), (C4×C4○D4).27C2, C4.199(C2×C4○D4), (C2×C4).1222(C2×D4), C2.35(C2×C8.C22), (C2×C4○D4).315C22, SmallGroup(128,1848)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.234D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=c3 >
Subgroups: 332 in 193 conjugacy classes, 90 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C42.6C4, SD16⋊C4, Q16⋊C4, D4.7D4, D4.D4, C4⋊2Q16, D4⋊2Q8, C4.Q16, C23.20D4, C4×C4○D4, C23.37C23, C42.234D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, C22.19C24, C2×C8.C22, D8⋊C22, C42.234D4
►On 64 points
(1 42 5 46)(2 64 6 60)(3 44 7 48)(4 58 8 62)(9 26 13 30)(10 18 14 22)(11 28 15 32)(12 20 16 24)(17 33 21 37)(19 35 23 39)(25 40 29 36)(27 34 31 38)(41 54 45 50)(43 56 47 52)(49 61 53 57)(51 63 55 59)
(1 27 55 22)(2 32 56 19)(3 29 49 24)(4 26 50 21)(5 31 51 18)(6 28 52 23)(7 25 53 20)(8 30 54 17)(9 45 33 62)(10 42 34 59)(11 47 35 64)(12 44 36 61)(13 41 37 58)(14 46 38 63)(15 43 39 60)(16 48 40 57)
(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 4 5 8)(2 7 6 3)(9 14 13 10)(11 12 15 16)(17 22 21 18)(19 20 23 24)(25 28 29 32)(26 31 30 27)(33 38 37 34)(35 36 39 40)(41 42 45 46)(43 48 47 44)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)
G:=sub<Sym(64)| (1,42,5,46)(2,64,6,60)(3,44,7,48)(4,58,8,62)(9,26,13,30)(10,18,14,22)(11,28,15,32)(12,20,16,24)(17,33,21,37)(19,35,23,39)(25,40,29,36)(27,34,31,38)(41,54,45,50)(43,56,47,52)(49,61,53,57)(51,63,55,59), (1,27,55,22)(2,32,56,19)(3,29,49,24)(4,26,50,21)(5,31,51,18)(6,28,52,23)(7,25,53,20)(8,30,54,17)(9,45,33,62)(10,42,34,59)(11,47,35,64)(12,44,36,61)(13,41,37,58)(14,46,38,63)(15,43,39,60)(16,48,40,57), (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), (1,4,5,8)(2,7,6,3)(9,14,13,10)(11,12,15,16)(17,22,21,18)(19,20,23,24)(25,28,29,32)(26,31,30,27)(33,38,37,34)(35,36,39,40)(41,42,45,46)(43,48,47,44)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63)>;
G:=Group( (1,42,5,46)(2,64,6,60)(3,44,7,48)(4,58,8,62)(9,26,13,30)(10,18,14,22)(11,28,15,32)(12,20,16,24)(17,33,21,37)(19,35,23,39)(25,40,29,36)(27,34,31,38)(41,54,45,50)(43,56,47,52)(49,61,53,57)(51,63,55,59), (1,27,55,22)(2,32,56,19)(3,29,49,24)(4,26,50,21)(5,31,51,18)(6,28,52,23)(7,25,53,20)(8,30,54,17)(9,45,33,62)(10,42,34,59)(11,47,35,64)(12,44,36,61)(13,41,37,58)(14,46,38,63)(15,43,39,60)(16,48,40,57), (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), (1,4,5,8)(2,7,6,3)(9,14,13,10)(11,12,15,16)(17,22,21,18)(19,20,23,24)(25,28,29,32)(26,31,30,27)(33,38,37,34)(35,36,39,40)(41,42,45,46)(43,48,47,44)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63) );
G=PermutationGroup([[(1,42,5,46),(2,64,6,60),(3,44,7,48),(4,58,8,62),(9,26,13,30),(10,18,14,22),(11,28,15,32),(12,20,16,24),(17,33,21,37),(19,35,23,39),(25,40,29,36),(27,34,31,38),(41,54,45,50),(43,56,47,52),(49,61,53,57),(51,63,55,59)], [(1,27,55,22),(2,32,56,19),(3,29,49,24),(4,26,50,21),(5,31,51,18),(6,28,52,23),(7,25,53,20),(8,30,54,17),(9,45,33,62),(10,42,34,59),(11,47,35,64),(12,44,36,61),(13,41,37,58),(14,46,38,63),(15,43,39,60),(16,48,40,57)], [(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)], [(1,4,5,8),(2,7,6,3),(9,14,13,10),(11,12,15,16),(17,22,21,18),(19,20,23,24),(25,28,29,32),(26,31,30,27),(33,38,37,34),(35,36,39,40),(41,42,45,46),(43,48,47,44),(49,56,53,52),(50,51,54,55),(57,64,61,60),(58,59,62,63)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4J | 4K | ··· | 4Q | 4R | 4S | 4T | 4U | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | C8.C22 | D8⋊C22 |
| kernel | C42.234D4 | C42.6C4 | SD16⋊C4 | Q16⋊C4 | D4.7D4 | D4.D4 | C4⋊2Q16 | D4⋊2Q8 | C4.Q16 | C23.20D4 | C4×C4○D4 | C23.37C23 | C42 | C22×C4 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.234D4 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 16 |
| 0 | 0 | 2 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 3 | 14 |
| 0 | 0 | 4 | 2 | 3 | 3 |
| 0 | 0 | 2 | 2 | 15 | 4 |
| 0 | 0 | 15 | 2 | 13 | 15 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 1 | 1 |
| 0 | 0 | 15 | 13 | 1 | 16 |
| 0 | 0 | 15 | 15 | 2 | 13 |
| 0 | 0 | 15 | 2 | 13 | 15 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,15,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,2,0,0,0,0,16,0,2,0,0,16,0,1,0,0,0,0,16,0,1],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,2,4,2,15,0,0,13,2,2,2,0,0,3,3,15,13,0,0,14,3,4,15],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,4,15,15,15,0,0,15,13,15,2,0,0,1,1,2,13,0,0,1,16,13,15] >;
C42.234D4 in GAP, Magma, Sage, TeX
C_4^2._{234}D_4 % in TeX
G:=Group("C4^2.234D4"); // GroupNames label
G:=SmallGroup(128,1848);
// by ID
G=gap.SmallGroup(128,1848);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,521,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations