p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.69D4, C42.150C23, (C4×D4).4C4, (C4×Q8).4C4, C22.5C4≀C2, C4⋊D4.11C4, C42.91(C2×C4), C22⋊Q8.11C4, (C22×C4).666D4, C8⋊C4.86C22, C42.6C4⋊36C2, C42.2C22⋊9C2, (C2×C42).194C22, C42.C2.97C22, C23.106(C22⋊C4), C42.C22⋊10C2, C4.4D4.116C22, C2.32(C42⋊C22), C23.36C23.11C2, C2.12(M4(2).8C22), C2.37(C2×C4≀C2), C4⋊C4.28(C2×C4), (C2×C8⋊C4)⋊16C2, (C2×D4).23(C2×C4), (C2×Q8).23(C2×C4), (C2×C4).1178(C2×D4), (C2×C4).144(C22×C4), (C22×C4).216(C2×C4), (C2×C4).322(C22⋊C4), C22.208(C2×C22⋊C4), SmallGroup(128,264)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.69D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=a2bc3 >
Subgroups: 212 in 109 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C42.C22, C42.2C22, C2×C8⋊C4, C42.6C4, C23.36C23, C42.69D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, M4(2).8C22, C2×C4≀C2, C42⋊C22, C42.69D4
►On 64 points
(1 44 5 48)(2 29 6 25)(3 46 7 42)(4 31 8 27)(9 23 13 19)(10 37 14 33)(11 17 15 21)(12 39 16 35)(18 53 22 49)(20 55 24 51)(26 63 30 59)(28 57 32 61)(34 56 38 52)(36 50 40 54)(41 62 45 58)(43 64 47 60)
(1 22 61 35)(2 19 62 40)(3 24 63 37)(4 21 64 34)(5 18 57 39)(6 23 58 36)(7 20 59 33)(8 17 60 38)(9 45 54 29)(10 42 55 26)(11 47 56 31)(12 44 49 28)(13 41 50 25)(14 46 51 30)(15 43 52 27)(16 48 53 32)
(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 17 22 60 61 38 35 8)(2 63 19 37 62 3 40 24)(4 5 21 18 64 57 34 39)(6 59 23 33 58 7 36 20)(9 51 45 30 54 14 29 46)(10 25 42 13 55 41 26 50)(11 16 47 48 56 53 31 32)(12 43 44 52 49 27 28 15)
G:=sub<Sym(64)| (1,44,5,48)(2,29,6,25)(3,46,7,42)(4,31,8,27)(9,23,13,19)(10,37,14,33)(11,17,15,21)(12,39,16,35)(18,53,22,49)(20,55,24,51)(26,63,30,59)(28,57,32,61)(34,56,38,52)(36,50,40,54)(41,62,45,58)(43,64,47,60), (1,22,61,35)(2,19,62,40)(3,24,63,37)(4,21,64,34)(5,18,57,39)(6,23,58,36)(7,20,59,33)(8,17,60,38)(9,45,54,29)(10,42,55,26)(11,47,56,31)(12,44,49,28)(13,41,50,25)(14,46,51,30)(15,43,52,27)(16,48,53,32), (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), (1,17,22,60,61,38,35,8)(2,63,19,37,62,3,40,24)(4,5,21,18,64,57,34,39)(6,59,23,33,58,7,36,20)(9,51,45,30,54,14,29,46)(10,25,42,13,55,41,26,50)(11,16,47,48,56,53,31,32)(12,43,44,52,49,27,28,15)>;
G:=Group( (1,44,5,48)(2,29,6,25)(3,46,7,42)(4,31,8,27)(9,23,13,19)(10,37,14,33)(11,17,15,21)(12,39,16,35)(18,53,22,49)(20,55,24,51)(26,63,30,59)(28,57,32,61)(34,56,38,52)(36,50,40,54)(41,62,45,58)(43,64,47,60), (1,22,61,35)(2,19,62,40)(3,24,63,37)(4,21,64,34)(5,18,57,39)(6,23,58,36)(7,20,59,33)(8,17,60,38)(9,45,54,29)(10,42,55,26)(11,47,56,31)(12,44,49,28)(13,41,50,25)(14,46,51,30)(15,43,52,27)(16,48,53,32), (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), (1,17,22,60,61,38,35,8)(2,63,19,37,62,3,40,24)(4,5,21,18,64,57,34,39)(6,59,23,33,58,7,36,20)(9,51,45,30,54,14,29,46)(10,25,42,13,55,41,26,50)(11,16,47,48,56,53,31,32)(12,43,44,52,49,27,28,15) );
G=PermutationGroup([[(1,44,5,48),(2,29,6,25),(3,46,7,42),(4,31,8,27),(9,23,13,19),(10,37,14,33),(11,17,15,21),(12,39,16,35),(18,53,22,49),(20,55,24,51),(26,63,30,59),(28,57,32,61),(34,56,38,52),(36,50,40,54),(41,62,45,58),(43,64,47,60)], [(1,22,61,35),(2,19,62,40),(3,24,63,37),(4,21,64,34),(5,18,57,39),(6,23,58,36),(7,20,59,33),(8,17,60,38),(9,45,54,29),(10,42,55,26),(11,47,56,31),(12,44,49,28),(13,41,50,25),(14,46,51,30),(15,43,52,27),(16,48,53,32)], [(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)], [(1,17,22,60,61,38,35,8),(2,63,19,37,62,3,40,24),(4,5,21,18,64,57,34,39),(6,59,23,33,58,7,36,20),(9,51,45,30,54,14,29,46),(10,25,42,13,55,41,26,50),(11,16,47,48,56,53,31,32),(12,43,44,52,49,27,28,15)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | M4(2).8C22 | C42⋊C22 |
| kernel | C42.69D4 | C42.C22 | C42.2C22 | C2×C8⋊C4 | C42.6C4 | C23.36C23 | C4×D4 | C4×Q8 | C4⋊D4 | C22⋊Q8 | C42 | C22×C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.69D4 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 16 | 0 | 13 | 0 |
| 0 | 0 | 16 | 1 | 13 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 1 | 13 | 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 |
| 12 | 3 | 0 | 0 | 0 | 0 |
| 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 9 | 8 |
| 0 | 0 | 0 | 2 | 0 | 8 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 15 |
| 12 | 3 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 9 | 8 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 9 | 8 | 2 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,16,16,0,0,0,0,0,1,0,0,15,16,13,13,0,0,0,1,0,0],[4,1,0,0,0,0,2,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[12,10,0,0,0,0,3,5,0,0,0,0,0,0,0,0,0,8,0,0,2,2,8,0,0,0,9,0,0,0,0,0,8,8,0,15],[12,10,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,8,8,0,0,9,9,0,2,0,0,8,0,0,0] >;
C42.69D4 in GAP, Magma, Sage, TeX
C_4^2._{69}D_4 % in TeX
G:=Group("C4^2.69D4"); // GroupNames label
G:=SmallGroup(128,264);
// by ID
G=gap.SmallGroup(128,264);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,520,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations