p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.324D4, (C4×C8).27C4, C8.20(C4⋊C4), (C2×C8).49Q8, (C2×C8).267D4, C4.46(C4⋊Q8), C4⋊1(C8.C4), C22.1(C4⋊Q8), (C22×C4).92Q8, C23.83(C2×Q8), C4.46(C4⋊1D4), C42.326(C2×C4), C2.9(C42⋊9C4), C4⋊M4(2).29C2, (C22×C8).550C22, (C22×C4).1347C23, (C2×C42).1062C22, (C2×M4(2)).169C22, (C2×C4×C8).45C2, C4.37(C2×C4⋊C4), (C2×C8).235(C2×C4), (C2×C4).732(C2×D4), (C2×C4).197(C2×Q8), (C2×C4).133(C4⋊C4), C2.13(C2×C8.C4), C22.106(C2×C4⋊C4), (C2×C8.C4).11C2, (C2×C4).546(C22×C4), SmallGroup(128,580)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.324D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 156 in 110 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C4×C8, C4⋊C8, C8.C4, C2×C42, C22×C8, C2×M4(2), C2×C4×C8, C4⋊M4(2), C2×C8.C4, C42.324D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C8.C4, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C42⋊9C4, C2×C8.C4, C42.324D4
►On 64 points
(1 40 27 10)(2 33 28 11)(3 34 29 12)(4 35 30 13)(5 36 31 14)(6 37 32 15)(7 38 25 16)(8 39 26 9)(17 49 46 62)(18 50 47 63)(19 51 48 64)(20 52 41 57)(21 53 42 58)(22 54 43 59)(23 55 44 60)(24 56 45 61)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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 49 3 55 5 53 7 51)(2 52 4 50 6 56 8 54)(9 43 11 41 13 47 15 45)(10 46 12 44 14 42 16 48)(17 34 23 36 21 38 19 40)(18 37 24 39 22 33 20 35)(25 64 27 62 29 60 31 58)(26 59 28 57 30 63 32 61)
G:=sub<Sym(64)| (1,40,27,10)(2,33,28,11)(3,34,29,12)(4,35,30,13)(5,36,31,14)(6,37,32,15)(7,38,25,16)(8,39,26,9)(17,49,46,62)(18,50,47,63)(19,51,48,64)(20,52,41,57)(21,53,42,58)(22,54,43,59)(23,55,44,60)(24,56,45,61), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,49,3,55,5,53,7,51)(2,52,4,50,6,56,8,54)(9,43,11,41,13,47,15,45)(10,46,12,44,14,42,16,48)(17,34,23,36,21,38,19,40)(18,37,24,39,22,33,20,35)(25,64,27,62,29,60,31,58)(26,59,28,57,30,63,32,61)>;
G:=Group( (1,40,27,10)(2,33,28,11)(3,34,29,12)(4,35,30,13)(5,36,31,14)(6,37,32,15)(7,38,25,16)(8,39,26,9)(17,49,46,62)(18,50,47,63)(19,51,48,64)(20,52,41,57)(21,53,42,58)(22,54,43,59)(23,55,44,60)(24,56,45,61), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,49,3,55,5,53,7,51)(2,52,4,50,6,56,8,54)(9,43,11,41,13,47,15,45)(10,46,12,44,14,42,16,48)(17,34,23,36,21,38,19,40)(18,37,24,39,22,33,20,35)(25,64,27,62,29,60,31,58)(26,59,28,57,30,63,32,61) );
G=PermutationGroup([[(1,40,27,10),(2,33,28,11),(3,34,29,12),(4,35,30,13),(5,36,31,14),(6,37,32,15),(7,38,25,16),(8,39,26,9),(17,49,46,62),(18,50,47,63),(19,51,48,64),(20,52,41,57),(21,53,42,58),(22,54,43,59),(23,55,44,60),(24,56,45,61)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,19,21,23),(18,20,22,24),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,49,3,55,5,53,7,51),(2,52,4,50,6,56,8,54),(9,43,11,41,13,47,15,45),(10,46,12,44,14,42,16,48),(17,34,23,36,21,38,19,40),(18,37,24,39,22,33,20,35),(25,64,27,62,29,60,31,58),(26,59,28,57,30,63,32,61)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | Q8 | C8.C4 |
| kernel | C42.324D4 | C2×C4×C8 | C4⋊M4(2) | C2×C8.C4 | C4×C8 | C42 | C2×C8 | C2×C8 | C22×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 8 | 2 | 4 | 4 | 2 | 16 |
Matrix representation of C42.324D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 15 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 |
| 0 | 0 | 13 | 15 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[0,2,0,0,15,0,0,0,0,0,2,13,0,0,5,15] >;
C42.324D4 in GAP, Magma, Sage, TeX
C_4^2._{324}D_4 % in TeX
G:=Group("C4^2.324D4"); // GroupNames label
G:=SmallGroup(128,580);
// by ID
G=gap.SmallGroup(128,580);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,100,2019,248,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations