p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.67D4, C42.148C23, C4.15C4≀C2, C4⋊Q8.16C4, C4⋊1D4.11C4, C42.89(C2×C4), (C22×C4).738D4, C42.6C4⋊35C2, C8⋊C4.146C22, C42.C22⋊9C2, (C2×C42).192C22, C22.2(C4.D4), C23.104(C22⋊C4), C4.4D4.115C22, C2.30(C42⋊C22), C22.26C24.10C2, C2.35(C2×C4≀C2), (C2×C4○D4).4C4, (C2×C8⋊C4)⋊15C2, (C2×D4).22(C2×C4), (C2×Q8).22(C2×C4), (C2×C4).1176(C2×D4), C2.12(C2×C4.D4), (C22×C4).214(C2×C4), (C2×C4).142(C22×C4), (C2×C4).320(C22⋊C4), C22.206(C2×C22⋊C4), SmallGroup(128,262)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.67D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 284 in 128 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×8], C8 [×6], C2×C4 [×6], C2×C4 [×11], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×4], C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8, C2×C4○D4 [×2], C42.C22 [×4], C2×C8⋊C4, C42.6C4, C22.26C24, C42.67D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C2×C4.D4, C2×C4≀C2, C42⋊C22, C42.67D4
►On 64 points
(1 45 19 34)(2 39 20 42)(3 47 21 36)(4 33 22 44)(5 41 23 38)(6 35 24 46)(7 43 17 40)(8 37 18 48)(9 56 63 30)(10 27 64 53)(11 50 57 32)(12 29 58 55)(13 52 59 26)(14 31 60 49)(15 54 61 28)(16 25 62 51)
(1 58 23 16)(2 63 24 13)(3 60 17 10)(4 57 18 15)(5 62 19 12)(6 59 20 9)(7 64 21 14)(8 61 22 11)(25 45 55 38)(26 42 56 35)(27 47 49 40)(28 44 50 37)(29 41 51 34)(30 46 52 39)(31 43 53 36)(32 48 54 33)
(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 37 58 28 23 44 16 50)(2 31 63 43 24 53 13 36)(3 46 60 52 17 39 10 30)(4 55 57 38 18 25 15 45)(5 33 62 32 19 48 12 54)(6 27 59 47 20 49 9 40)(7 42 64 56 21 35 14 26)(8 51 61 34 22 29 11 41)
G:=sub<Sym(64)| (1,45,19,34)(2,39,20,42)(3,47,21,36)(4,33,22,44)(5,41,23,38)(6,35,24,46)(7,43,17,40)(8,37,18,48)(9,56,63,30)(10,27,64,53)(11,50,57,32)(12,29,58,55)(13,52,59,26)(14,31,60,49)(15,54,61,28)(16,25,62,51), (1,58,23,16)(2,63,24,13)(3,60,17,10)(4,57,18,15)(5,62,19,12)(6,59,20,9)(7,64,21,14)(8,61,22,11)(25,45,55,38)(26,42,56,35)(27,47,49,40)(28,44,50,37)(29,41,51,34)(30,46,52,39)(31,43,53,36)(32,48,54,33), (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), (1,37,58,28,23,44,16,50)(2,31,63,43,24,53,13,36)(3,46,60,52,17,39,10,30)(4,55,57,38,18,25,15,45)(5,33,62,32,19,48,12,54)(6,27,59,47,20,49,9,40)(7,42,64,56,21,35,14,26)(8,51,61,34,22,29,11,41)>;
G:=Group( (1,45,19,34)(2,39,20,42)(3,47,21,36)(4,33,22,44)(5,41,23,38)(6,35,24,46)(7,43,17,40)(8,37,18,48)(9,56,63,30)(10,27,64,53)(11,50,57,32)(12,29,58,55)(13,52,59,26)(14,31,60,49)(15,54,61,28)(16,25,62,51), (1,58,23,16)(2,63,24,13)(3,60,17,10)(4,57,18,15)(5,62,19,12)(6,59,20,9)(7,64,21,14)(8,61,22,11)(25,45,55,38)(26,42,56,35)(27,47,49,40)(28,44,50,37)(29,41,51,34)(30,46,52,39)(31,43,53,36)(32,48,54,33), (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), (1,37,58,28,23,44,16,50)(2,31,63,43,24,53,13,36)(3,46,60,52,17,39,10,30)(4,55,57,38,18,25,15,45)(5,33,62,32,19,48,12,54)(6,27,59,47,20,49,9,40)(7,42,64,56,21,35,14,26)(8,51,61,34,22,29,11,41) );
G=PermutationGroup([(1,45,19,34),(2,39,20,42),(3,47,21,36),(4,33,22,44),(5,41,23,38),(6,35,24,46),(7,43,17,40),(8,37,18,48),(9,56,63,30),(10,27,64,53),(11,50,57,32),(12,29,58,55),(13,52,59,26),(14,31,60,49),(15,54,61,28),(16,25,62,51)], [(1,58,23,16),(2,63,24,13),(3,60,17,10),(4,57,18,15),(5,62,19,12),(6,59,20,9),(7,64,21,14),(8,61,22,11),(25,45,55,38),(26,42,56,35),(27,47,49,40),(28,44,50,37),(29,41,51,34),(30,46,52,39),(31,43,53,36),(32,48,54,33)], [(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)], [(1,37,58,28,23,44,16,50),(2,31,63,43,24,53,13,36),(3,46,60,52,17,39,10,30),(4,55,57,38,18,25,15,45),(5,33,62,32,19,48,12,54),(6,27,59,47,20,49,9,40),(7,42,64,56,21,35,14,26),(8,51,61,34,22,29,11,41)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | C4.D4 | C42⋊C22 |
| kernel | C42.67D4 | C42.C22 | C2×C8⋊C4 | C42.6C4 | C22.26C24 | C4⋊1D4 | C4⋊Q8 | C2×C4○D4 | C42 | C22×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.67D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 4 | 16 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 10 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 15 |
| 0 | 0 | 5 | 3 | 15 | 15 |
| 0 | 0 | 5 | 5 | 0 | 15 |
| 0 | 0 | 12 | 7 | 14 | 11 |
| 11 | 7 | 0 | 0 | 0 | 0 |
| 7 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 8 |
| 0 | 0 | 12 | 3 | 9 | 9 |
| 0 | 0 | 12 | 12 | 0 | 9 |
| 0 | 0 | 5 | 0 | 14 | 0 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,16,0,0,0,0,4,0,0,0,0,1,16,0,0,0,0,0,15,0,13],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[6,10,0,0,0,0,7,11,0,0,0,0,0,0,3,5,5,12,0,0,3,3,5,7,0,0,0,15,0,14,0,0,15,15,15,11],[11,7,0,0,0,0,7,11,0,0,0,0,0,0,14,12,12,5,0,0,14,3,12,0,0,0,0,9,0,14,0,0,8,9,9,0] >;
C42.67D4 in GAP, Magma, Sage, TeX
C_4^2._{67}D_4 % in TeX
G:=Group("C4^2.67D4"); // GroupNames label
G:=SmallGroup(128,262);
// by ID
G=gap.SmallGroup(128,262);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,352,1123,1018,248,1971,102]);
// 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*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations