p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.19Q8, C42.377D4, C42.629C23, C4⋊C8.14C4, C8⋊1C8⋊28C2, C8⋊2C8⋊27C2, C4.45(C8○D4), C22⋊C8.12C4, C4.124(C4○D8), (C22×C4).12Q8, C4⋊C8.269C22, C23.20(C4⋊C4), C42.125(C2×C4), (C4×C8).237C22, (C22×C4).250D4, C42.6C4.30C2, (C2×C42).228C22, C2.7(M4(2).C4), C42.12C4.30C2, C2.6(C23.25D4), C2.7(C42.6C22), (C2×C8).28(C2×C4), (C2×C4).36(C4⋊C4), C22.86(C2×C4⋊C4), (C2×C4).156(C2×Q8), (C2×C4).1465(C2×D4), (C2×C4).511(C22×C4), (C22×C4).250(C2×C4), SmallGroup(128,304)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.Q8
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, cac-1=a-1b2, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 116 in 75 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C8⋊2C8, C8⋊1C8, C42.12C4, C42.6C4, C42.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C8○D4, C4○D8, C42.6C22, C23.25D4, M4(2).C4, C42.Q8
►On 64 points
(1 7 5 3)(2 41 6 45)(4 43 8 47)(9 19 13 23)(10 12 14 16)(11 21 15 17)(18 20 22 24)(25 58 29 62)(26 28 30 32)(27 60 31 64)(33 39 37 35)(34 56 38 52)(36 50 40 54)(42 48 46 44)(49 55 53 51)(57 59 61 63)
(1 33 46 53)(2 34 47 54)(3 35 48 55)(4 36 41 56)(5 37 42 49)(6 38 43 50)(7 39 44 51)(8 40 45 52)(9 64 21 29)(10 57 22 30)(11 58 23 31)(12 59 24 32)(13 60 17 25)(14 61 18 26)(15 62 19 27)(16 63 20 28)
(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 32 39 10 42 63 55 18)(2 27 40 13 43 58 56 21)(3 30 33 16 44 61 49 24)(4 25 34 11 45 64 50 19)(5 28 35 14 46 59 51 22)(6 31 36 9 47 62 52 17)(7 26 37 12 48 57 53 20)(8 29 38 15 41 60 54 23)
G:=sub<Sym(64)| (1,7,5,3)(2,41,6,45)(4,43,8,47)(9,19,13,23)(10,12,14,16)(11,21,15,17)(18,20,22,24)(25,58,29,62)(26,28,30,32)(27,60,31,64)(33,39,37,35)(34,56,38,52)(36,50,40,54)(42,48,46,44)(49,55,53,51)(57,59,61,63), (1,33,46,53)(2,34,47,54)(3,35,48,55)(4,36,41,56)(5,37,42,49)(6,38,43,50)(7,39,44,51)(8,40,45,52)(9,64,21,29)(10,57,22,30)(11,58,23,31)(12,59,24,32)(13,60,17,25)(14,61,18,26)(15,62,19,27)(16,63,20,28), (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), (1,32,39,10,42,63,55,18)(2,27,40,13,43,58,56,21)(3,30,33,16,44,61,49,24)(4,25,34,11,45,64,50,19)(5,28,35,14,46,59,51,22)(6,31,36,9,47,62,52,17)(7,26,37,12,48,57,53,20)(8,29,38,15,41,60,54,23)>;
G:=Group( (1,7,5,3)(2,41,6,45)(4,43,8,47)(9,19,13,23)(10,12,14,16)(11,21,15,17)(18,20,22,24)(25,58,29,62)(26,28,30,32)(27,60,31,64)(33,39,37,35)(34,56,38,52)(36,50,40,54)(42,48,46,44)(49,55,53,51)(57,59,61,63), (1,33,46,53)(2,34,47,54)(3,35,48,55)(4,36,41,56)(5,37,42,49)(6,38,43,50)(7,39,44,51)(8,40,45,52)(9,64,21,29)(10,57,22,30)(11,58,23,31)(12,59,24,32)(13,60,17,25)(14,61,18,26)(15,62,19,27)(16,63,20,28), (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), (1,32,39,10,42,63,55,18)(2,27,40,13,43,58,56,21)(3,30,33,16,44,61,49,24)(4,25,34,11,45,64,50,19)(5,28,35,14,46,59,51,22)(6,31,36,9,47,62,52,17)(7,26,37,12,48,57,53,20)(8,29,38,15,41,60,54,23) );
G=PermutationGroup([[(1,7,5,3),(2,41,6,45),(4,43,8,47),(9,19,13,23),(10,12,14,16),(11,21,15,17),(18,20,22,24),(25,58,29,62),(26,28,30,32),(27,60,31,64),(33,39,37,35),(34,56,38,52),(36,50,40,54),(42,48,46,44),(49,55,53,51),(57,59,61,63)], [(1,33,46,53),(2,34,47,54),(3,35,48,55),(4,36,41,56),(5,37,42,49),(6,38,43,50),(7,39,44,51),(8,40,45,52),(9,64,21,29),(10,57,22,30),(11,58,23,31),(12,59,24,32),(13,60,17,25),(14,61,18,26),(15,62,19,27),(16,63,20,28)], [(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)], [(1,32,39,10,42,63,55,18),(2,27,40,13,43,58,56,21),(3,30,33,16,44,61,49,24),(4,25,34,11,45,64,50,19),(5,28,35,14,46,59,51,22),(6,31,36,9,47,62,52,17),(7,26,37,12,48,57,53,20),(8,29,38,15,41,60,54,23)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 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 | C4 | C4 | D4 | Q8 | D4 | Q8 | C8○D4 | C4○D8 | M4(2).C4 |
| kernel | C42.Q8 | C8⋊2C8 | C8⋊1C8 | C42.12C4 | C42.6C4 | C22⋊C8 | C4⋊C8 | C42 | C42 | C22×C4 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 8 | 2 |
Matrix representation of C42.Q8 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 1 |
| 4 | 0 | 0 | 0 |
| 11 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 8 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 7 | 15 | 0 | 0 |
| 8 | 10 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 9 | 8 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,16,0,0,0,1],[4,11,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[8,4,0,0,0,2,0,0,0,0,1,1,0,0,15,16],[7,8,0,0,15,10,0,0,0,0,9,9,0,0,0,8] >;
C42.Q8 in GAP, Magma, Sage, TeX
C_4^2.Q_8
% in TeX
G:=Group("C4^2.Q8"); // GroupNames label
G:=SmallGroup(128,304);
// by ID
G=gap.SmallGroup(128,304);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,723,184,1123,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b*c^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations