p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.430D4, M4(2).4Q8, M4(2).26D4, C4⋊C8.16C4, C4.38(C4×Q8), C4.146(C4×D4), C4⋊2(C8.C4), C22.6(C4⋊Q8), C23.86(C2×Q8), (C22×C4).52Q8, C42.153(C2×C4), C4.201(C4⋊D4), (C4×M4(2)).24C2, C4.C42.3C2, C4.119(C22⋊Q8), C4⋊M4(2).31C2, (C2×C42).313C22, (C22×C8).167C22, C22.8(C42.C2), (C22×C4).1399C23, C2.12(M4(2).C4), (C2×M4(2)).204C22, C2.17(C23.65C23), (C2×C4⋊C8).45C2, (C2×C8).45(C2×C4), (C2×C4).56(C4⋊C4), (C2×C4).274(C2×Q8), (C2×C4).1542(C2×D4), C2.15(C2×C8.C4), C22.117(C2×C4⋊C4), (C2×C8.C4).15C2, (C2×C4).763(C4○D4), (C2×C4).555(C22×C4), SmallGroup(128,682)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.430D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 148 in 100 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4⋊C8, C8.C4, C2×C42, C22×C8, C2×M4(2), C2×M4(2), C4.C42, C4×M4(2), C2×C4⋊C8, C4⋊M4(2), C2×C8.C4, C42.430D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, C2×C8.C4, M4(2).C4, C42.430D4
►On 64 points
(1 54 59 22)(2 23 60 55)(3 56 61 24)(4 17 62 49)(5 50 63 18)(6 19 64 51)(7 52 57 20)(8 21 58 53)(9 41 34 31)(10 32 35 42)(11 43 36 25)(12 26 37 44)(13 45 38 27)(14 28 39 46)(15 47 40 29)(16 30 33 48)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(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 30 7 32 5 26 3 28)(2 25 8 27 6 29 4 31)(9 55 11 53 13 51 15 49)(10 50 12 56 14 54 16 52)(17 34 23 36 21 38 19 40)(18 37 24 39 22 33 20 35)(41 60 43 58 45 64 47 62)(42 63 44 61 46 59 48 57)
G:=sub<Sym(64)| (1,54,59,22)(2,23,60,55)(3,56,61,24)(4,17,62,49)(5,50,63,18)(6,19,64,51)(7,52,57,20)(8,21,58,53)(9,41,34,31)(10,32,35,42)(11,43,36,25)(12,26,37,44)(13,45,38,27)(14,28,39,46)(15,47,40,29)(16,30,33,48), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (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), (1,30,7,32,5,26,3,28)(2,25,8,27,6,29,4,31)(9,55,11,53,13,51,15,49)(10,50,12,56,14,54,16,52)(17,34,23,36,21,38,19,40)(18,37,24,39,22,33,20,35)(41,60,43,58,45,64,47,62)(42,63,44,61,46,59,48,57)>;
G:=Group( (1,54,59,22)(2,23,60,55)(3,56,61,24)(4,17,62,49)(5,50,63,18)(6,19,64,51)(7,52,57,20)(8,21,58,53)(9,41,34,31)(10,32,35,42)(11,43,36,25)(12,26,37,44)(13,45,38,27)(14,28,39,46)(15,47,40,29)(16,30,33,48), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (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), (1,30,7,32,5,26,3,28)(2,25,8,27,6,29,4,31)(9,55,11,53,13,51,15,49)(10,50,12,56,14,54,16,52)(17,34,23,36,21,38,19,40)(18,37,24,39,22,33,20,35)(41,60,43,58,45,64,47,62)(42,63,44,61,46,59,48,57) );
G=PermutationGroup([[(1,54,59,22),(2,23,60,55),(3,56,61,24),(4,17,62,49),(5,50,63,18),(6,19,64,51),(7,52,57,20),(8,21,58,53),(9,41,34,31),(10,32,35,42),(11,43,36,25),(12,26,37,44),(13,45,38,27),(14,28,39,46),(15,47,40,29),(16,30,33,48)], [(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,23,21,19),(18,24,22,20),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(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)], [(1,30,7,32,5,26,3,28),(2,25,8,27,6,29,4,31),(9,55,11,53,13,51,15,49),(10,50,12,56,14,54,16,52),(17,34,23,36,21,38,19,40),(18,37,24,39,22,33,20,35),(41,60,43,58,45,64,47,62),(42,63,44,61,46,59,48,57)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 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 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | Q8 | C4○D4 | C8.C4 | M4(2).C4 |
| kernel | C42.430D4 | C4.C42 | C4×M4(2) | C2×C4⋊C8 | C4⋊M4(2) | C2×C8.C4 | C4⋊C8 | C42 | M4(2) | M4(2) | C22×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 2 |
Matrix representation of C42.430D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 11 | 13 |
| 0 | 8 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 13 | 6 |
| 0 | 0 | 6 | 4 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,15,0,0,0,0,4,11,0,0,11,13],[0,9,0,0,8,0,0,0,0,0,13,6,0,0,6,4] >;
C42.430D4 in GAP, Magma, Sage, TeX
C_4^2._{430}D_4 % in TeX
G:=Group("C4^2.430D4"); // GroupNames label
G:=SmallGroup(128,682);
// by ID
G=gap.SmallGroup(128,682);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2019,1018,248,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations