p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.224D4, C42.340C23, (C4×Q16)⋊4C2, (C2×C4)⋊11Q16, C4.46(C2×Q16), C4○2(C4.Q16), C4.Q16⋊49C2, C4○2(C4⋊2Q16), C4⋊2Q16⋊48C2, C4⋊C4.57C23, (C4×C8).70C22, Q8.2(C4○D4), C2.9(C22×Q16), C4⋊C8.284C22, (C2×C4).302C24, (C2×C8).147C23, C4○2(C22⋊Q16), C23.669(C2×D4), (C22×C4).804D4, C4⋊Q8.264C22, C22⋊Q16.5C2, (C4×Q8).70C22, C22.16(C2×Q16), (C2×Q8).373C23, C2.D8.170C22, C4○2(C23.48D4), C22⋊C8.174C22, (C2×C42).829C22, (C2×Q16).120C22, C23.48D4.5C2, C22.562(C22×D4), C22⋊Q8.165C22, C2.29(D8⋊C22), C42.12C4.34C2, (C22×C4).1018C23, Q8⋊C4.152C22, (C22×Q8).476C22, C2.103(C22.19C24), C23.37C23.28C2, (C2×C4×Q8).53C2, C4.187(C2×C4○D4), (C2×C4).1582(C2×D4), (C2×C4⋊C4).933C22, SmallGroup(128,1836)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.224D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=a2c3 >
Subgroups: 308 in 192 conjugacy classes, 98 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×Q16, C22×Q8, C42.12C4, C4×Q16, C22⋊Q16, C4⋊2Q16, C4.Q16, C23.48D4, C2×C4×Q8, C23.37C23, C42.224D4
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C24, C2×Q16, C22×D4, C2×C4○D4, C22.19C24, C22×Q16, D8⋊C22, C42.224D4
►On 64 points
(1 7 5 3)(2 58 6 62)(4 60 8 64)(9 15 13 11)(10 35 14 39)(12 37 16 33)(17 23 21 19)(18 53 22 49)(20 55 24 51)(25 48 29 44)(26 32 30 28)(27 42 31 46)(34 40 38 36)(41 47 45 43)(50 56 54 52)(57 63 61 59)
(1 52 63 19)(2 53 64 20)(3 54 57 21)(4 55 58 22)(5 56 59 23)(6 49 60 24)(7 50 61 17)(8 51 62 18)(9 47 40 26)(10 48 33 27)(11 41 34 28)(12 42 35 29)(13 43 36 30)(14 44 37 31)(15 45 38 32)(16 46 39 25)
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)
G:=sub<Sym(64)| (1,7,5,3)(2,58,6,62)(4,60,8,64)(9,15,13,11)(10,35,14,39)(12,37,16,33)(17,23,21,19)(18,53,22,49)(20,55,24,51)(25,48,29,44)(26,32,30,28)(27,42,31,46)(34,40,38,36)(41,47,45,43)(50,56,54,52)(57,63,61,59), (1,52,63,19)(2,53,64,20)(3,54,57,21)(4,55,58,22)(5,56,59,23)(6,49,60,24)(7,50,61,17)(8,51,62,18)(9,47,40,26)(10,48,33,27)(11,41,34,28)(12,42,35,29)(13,43,36,30)(14,44,37,31)(15,45,38,32)(16,46,39,25), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57)>;
G:=Group( (1,7,5,3)(2,58,6,62)(4,60,8,64)(9,15,13,11)(10,35,14,39)(12,37,16,33)(17,23,21,19)(18,53,22,49)(20,55,24,51)(25,48,29,44)(26,32,30,28)(27,42,31,46)(34,40,38,36)(41,47,45,43)(50,56,54,52)(57,63,61,59), (1,52,63,19)(2,53,64,20)(3,54,57,21)(4,55,58,22)(5,56,59,23)(6,49,60,24)(7,50,61,17)(8,51,62,18)(9,47,40,26)(10,48,33,27)(11,41,34,28)(12,42,35,29)(13,43,36,30)(14,44,37,31)(15,45,38,32)(16,46,39,25), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57) );
G=PermutationGroup([[(1,7,5,3),(2,58,6,62),(4,60,8,64),(9,15,13,11),(10,35,14,39),(12,37,16,33),(17,23,21,19),(18,53,22,49),(20,55,24,51),(25,48,29,44),(26,32,30,28),(27,42,31,46),(34,40,38,36),(41,47,45,43),(50,56,54,52),(57,63,61,59)], [(1,52,63,19),(2,53,64,20),(3,54,57,21),(4,55,58,22),(5,56,59,23),(6,49,60,24),(7,50,61,17),(8,51,62,18),(9,47,40,26),(10,48,33,27),(11,41,34,28),(12,42,35,29),(13,43,36,30),(14,44,37,31),(15,45,38,32),(16,46,39,25)], [(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)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,51,13,55),(10,50,14,54),(11,49,15,53),(12,56,16,52),(17,37,21,33),(18,36,22,40),(19,35,23,39),(20,34,24,38),(41,64,45,60),(42,63,46,59),(43,62,47,58),(44,61,48,57)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 4U | 4V | 4W | 4X | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q16 | C4○D4 | D8⋊C22 |
| kernel | C42.224D4 | C42.12C4 | C4×Q16 | C22⋊Q16 | C4⋊2Q16 | C4.Q16 | C23.48D4 | C2×C4×Q8 | C23.37C23 | C42 | C22×C4 | C2×C4 | Q8 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.224D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 14 | 3 |
| 0 | 0 | 14 | 14 |
| 16 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(17))| [16,2,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,0,1,0,0,16,0],[1,15,0,0,1,16,0,0,0,0,14,14,0,0,3,14],[16,0,0,0,16,1,0,0,0,0,13,0,0,0,0,4] >;
C42.224D4 in GAP, Magma, Sage, TeX
C_4^2._{224}D_4 % in TeX
G:=Group("C4^2.224D4"); // GroupNames label
G:=SmallGroup(128,1836);
// by ID
G=gap.SmallGroup(128,1836);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,80,4037,1027,124]);
// 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=a^-1*b^2,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations