p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.315D4, C42.613C23, C4.11C4≀C2, D4⋊C8⋊30C2, (C2×C4).71D8, C4.81(C2×D8), C4⋊Q8.10C4, C4.1(C8○D4), C4⋊D4.5C4, C4⋊1D4.8C4, (C2×C4).67SD16, C4.95(C2×SD16), (C4×D4).5C22, C4⋊C8.197C22, (C4×C8).363C22, C42.257(C2×C4), (C22×C4).572D4, C4.19(D4⋊C4), C4⋊M4(2)⋊17C2, C22.5(D4⋊C4), C23.97(C22⋊C4), (C2×C42).1035C22, C22.26C24.3C2, (C2×C4×C8)⋊3C2, C2.12(C2×C4≀C2), C4⋊C4.53(C2×C4), (C2×D4).52(C2×C4), C2.6(C2×D4⋊C4), (C2×C4).1141(C2×D4), (C22×C4).396(C2×C4), (C2×C4).318(C22×C4), (C2×C4).167(C22⋊C4), C22.168(C2×C22⋊C4), C2.18((C22×C8)⋊C2), SmallGroup(128,224)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.315D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 292 in 138 conjugacy classes, 56 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×C4○D4, D4⋊C8, C2×C4×C8, C4⋊M4(2), C22.26C24, C42.315D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C4≀C2, C2×C22⋊C4, C8○D4, C2×D8, C2×SD16, (C22×C8)⋊C2, C2×D4⋊C4, C2×C4≀C2, C42.315D4
►On 64 points
(1 9 51 32)(2 10 52 25)(3 11 53 26)(4 12 54 27)(5 13 55 28)(6 14 56 29)(7 15 49 30)(8 16 50 31)(17 48 60 40)(18 41 61 33)(19 42 62 34)(20 43 63 35)(21 44 64 36)(22 45 57 37)(23 46 58 38)(24 47 59 39)
(1 22 55 61)(2 23 56 62)(3 24 49 63)(4 17 50 64)(5 18 51 57)(6 19 52 58)(7 20 53 59)(8 21 54 60)(9 45 28 33)(10 46 29 34)(11 47 30 35)(12 48 31 36)(13 41 32 37)(14 42 25 38)(15 43 26 39)(16 44 27 40)
(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 21 22 54 55 60 61 8)(2 53 23 59 56 7 62 20)(3 58 24 6 49 19 63 52)(4 5 17 18 50 51 64 57)(9 48 45 31 28 36 33 12)(10 30 46 35 29 11 34 47)(13 44 41 27 32 40 37 16)(14 26 42 39 25 15 38 43)
G:=sub<Sym(64)| (1,9,51,32)(2,10,52,25)(3,11,53,26)(4,12,54,27)(5,13,55,28)(6,14,56,29)(7,15,49,30)(8,16,50,31)(17,48,60,40)(18,41,61,33)(19,42,62,34)(20,43,63,35)(21,44,64,36)(22,45,57,37)(23,46,58,38)(24,47,59,39), (1,22,55,61)(2,23,56,62)(3,24,49,63)(4,17,50,64)(5,18,51,57)(6,19,52,58)(7,20,53,59)(8,21,54,60)(9,45,28,33)(10,46,29,34)(11,47,30,35)(12,48,31,36)(13,41,32,37)(14,42,25,38)(15,43,26,39)(16,44,27,40), (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), (1,21,22,54,55,60,61,8)(2,53,23,59,56,7,62,20)(3,58,24,6,49,19,63,52)(4,5,17,18,50,51,64,57)(9,48,45,31,28,36,33,12)(10,30,46,35,29,11,34,47)(13,44,41,27,32,40,37,16)(14,26,42,39,25,15,38,43)>;
G:=Group( (1,9,51,32)(2,10,52,25)(3,11,53,26)(4,12,54,27)(5,13,55,28)(6,14,56,29)(7,15,49,30)(8,16,50,31)(17,48,60,40)(18,41,61,33)(19,42,62,34)(20,43,63,35)(21,44,64,36)(22,45,57,37)(23,46,58,38)(24,47,59,39), (1,22,55,61)(2,23,56,62)(3,24,49,63)(4,17,50,64)(5,18,51,57)(6,19,52,58)(7,20,53,59)(8,21,54,60)(9,45,28,33)(10,46,29,34)(11,47,30,35)(12,48,31,36)(13,41,32,37)(14,42,25,38)(15,43,26,39)(16,44,27,40), (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), (1,21,22,54,55,60,61,8)(2,53,23,59,56,7,62,20)(3,58,24,6,49,19,63,52)(4,5,17,18,50,51,64,57)(9,48,45,31,28,36,33,12)(10,30,46,35,29,11,34,47)(13,44,41,27,32,40,37,16)(14,26,42,39,25,15,38,43) );
G=PermutationGroup([[(1,9,51,32),(2,10,52,25),(3,11,53,26),(4,12,54,27),(5,13,55,28),(6,14,56,29),(7,15,49,30),(8,16,50,31),(17,48,60,40),(18,41,61,33),(19,42,62,34),(20,43,63,35),(21,44,64,36),(22,45,57,37),(23,46,58,38),(24,47,59,39)], [(1,22,55,61),(2,23,56,62),(3,24,49,63),(4,17,50,64),(5,18,51,57),(6,19,52,58),(7,20,53,59),(8,21,54,60),(9,45,28,33),(10,46,29,34),(11,47,30,35),(12,48,31,36),(13,41,32,37),(14,42,25,38),(15,43,26,39),(16,44,27,40)], [(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)], [(1,21,22,54,55,60,61,8),(2,53,23,59,56,7,62,20),(3,58,24,6,49,19,63,52),(4,5,17,18,50,51,64,57),(9,48,45,31,28,36,33,12),(10,30,46,35,29,11,34,47),(13,44,41,27,32,40,37,16),(14,26,42,39,25,15,38,43)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | SD16 | C4≀C2 | C8○D4 |
| kernel | C42.315D4 | D4⋊C8 | C2×C4×C8 | C4⋊M4(2) | C22.26C24 | C4⋊D4 | C4⋊1D4 | C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C42.315D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 12 | 12 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 5 | 10 |
| 12 | 12 | 0 | 0 |
| 11 | 5 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,1,0,0,15,1],[4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[12,11,0,0,12,0,0,0,0,0,0,5,0,0,7,10],[12,11,0,0,12,5,0,0,0,0,0,12,0,0,7,0] >;
C42.315D4 in GAP, Magma, Sage, TeX
C_4^2._{315}D_4 % in TeX
G:=Group("C4^2.315D4"); // GroupNames label
G:=SmallGroup(128,224);
// by ID
G=gap.SmallGroup(128,224);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,723,1123,570,136,172]);
// 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,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations