direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8×Q8, C42.75C22, C8○3(C4⋊C4), C8○2(C4⋊C8), C4.4(C2×C8), (C4×C8).3C2, C4⋊C4.12C4, C4⋊C8.11C2, C2.2(C4×Q8), (C2×Q8).9C4, C4.23(C2×Q8), C2.3(C8○D4), C2.5(C22×C8), (C4×Q8).11C2, C4.54(C4○D4), (C2×C8).65C22, (C2×C4).156C23, C22.24(C22×C4), (C2×C8)○(C4×Q8), (C2×C4).47(C2×C4), SmallGroup(64,126)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×Q8
G = < a,b,c | a8=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C8×Q8
►Regular action on 64 points
(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 39 19 30)(2 40 20 31)(3 33 21 32)(4 34 22 25)(5 35 23 26)(6 36 24 27)(7 37 17 28)(8 38 18 29)(9 44 64 53)(10 45 57 54)(11 46 58 55)(12 47 59 56)(13 48 60 49)(14 41 61 50)(15 42 62 51)(16 43 63 52)
(1 54 19 45)(2 55 20 46)(3 56 21 47)(4 49 22 48)(5 50 23 41)(6 51 24 42)(7 52 17 43)(8 53 18 44)(9 38 64 29)(10 39 57 30)(11 40 58 31)(12 33 59 32)(13 34 60 25)(14 35 61 26)(15 36 62 27)(16 37 63 28)
G:=sub<Sym(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), (1,39,19,30)(2,40,20,31)(3,33,21,32)(4,34,22,25)(5,35,23,26)(6,36,24,27)(7,37,17,28)(8,38,18,29)(9,44,64,53)(10,45,57,54)(11,46,58,55)(12,47,59,56)(13,48,60,49)(14,41,61,50)(15,42,62,51)(16,43,63,52), (1,54,19,45)(2,55,20,46)(3,56,21,47)(4,49,22,48)(5,50,23,41)(6,51,24,42)(7,52,17,43)(8,53,18,44)(9,38,64,29)(10,39,57,30)(11,40,58,31)(12,33,59,32)(13,34,60,25)(14,35,61,26)(15,36,62,27)(16,37,63,28)>;
G:=Group( (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), (1,39,19,30)(2,40,20,31)(3,33,21,32)(4,34,22,25)(5,35,23,26)(6,36,24,27)(7,37,17,28)(8,38,18,29)(9,44,64,53)(10,45,57,54)(11,46,58,55)(12,47,59,56)(13,48,60,49)(14,41,61,50)(15,42,62,51)(16,43,63,52), (1,54,19,45)(2,55,20,46)(3,56,21,47)(4,49,22,48)(5,50,23,41)(6,51,24,42)(7,52,17,43)(8,53,18,44)(9,38,64,29)(10,39,57,30)(11,40,58,31)(12,33,59,32)(13,34,60,25)(14,35,61,26)(15,36,62,27)(16,37,63,28) );
G=PermutationGroup([[(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)], [(1,39,19,30),(2,40,20,31),(3,33,21,32),(4,34,22,25),(5,35,23,26),(6,36,24,27),(7,37,17,28),(8,38,18,29),(9,44,64,53),(10,45,57,54),(11,46,58,55),(12,47,59,56),(13,48,60,49),(14,41,61,50),(15,42,62,51),(16,43,63,52)], [(1,54,19,45),(2,55,20,46),(3,56,21,47),(4,49,22,48),(5,50,23,41),(6,51,24,42),(7,52,17,43),(8,53,18,44),(9,38,64,29),(10,39,57,30),(11,40,58,31),(12,33,59,32),(13,34,60,25),(14,35,61,26),(15,36,62,27),(16,37,63,28)]])
C8×Q8 is a maximal subgroup of
Q8⋊C16 SD16⋊C8 C8⋊15SD16 C8⋊9Q16 Q8.M4(2) Q8⋊2M4(2) C8⋊14SD16 C8⋊13SD16 Q8⋊1Q16 C8⋊8Q16 C8⋊7Q16 Q8.1Q16 Q8.2SD16 Q8.3SD16 Q8.2D8 Q8.2Q16 C16⋊4Q8 C42.286C23 M4(2)⋊9Q8 C42.291C23 C42.294C23 Q8⋊6M4(2) C42.695C23 Q8.4M4(2) C42.696C23 C42.304C23 C42.697C23 Q8⋊7M4(2) C42.308C23 C42.309C23 Q8⋊4D8 Q8⋊7SD16 C42.501C23 C42.502C23 Q8⋊8SD16 Q8⋊5Q16 C42.505C23 C42.506C23 D8⋊6Q8 SD16⋊4Q8 Q16⋊6Q8 Q8⋊5D8 Q8⋊9SD16 C42.527C23 C42.528C23 Q8⋊6Q16 C42.530C23
C4p.(C2×C8): Q16⋊5C8 Dic6⋊C8 Dic10⋊5C8 Dic10⋊C8 Dic14⋊C8 ...
C8×Q8 is a maximal quotient of
C4⋊C4⋊3C8 C16⋊4Q8 Dic10⋊C8
C42.D2p: C42.61Q8 C42.327D4 Dic6⋊C8 Dic10⋊5C8 Dic14⋊C8 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | Q8 | C4○D4 | C8○D4 |
| kernel | C8×Q8 | C4×C8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | Q8 | C8 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 16 | 2 | 2 | 4 |
Matrix representation of C8×Q8 ►in GL3(𝔽17) generated by
| 8 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 16 | 2 |
| 0 | 16 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 7 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(17))| [8,0,0,0,1,0,0,0,1],[1,0,0,0,16,16,0,2,1],[1,0,0,0,0,12,0,7,0] >;
C8×Q8 in GAP, Magma, Sage, TeX
C_8\times Q_8
% in TeX
G:=Group("C8xQ8"); // GroupNames label
G:=SmallGroup(64,126);
// by ID
G=gap.SmallGroup(64,126);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,122,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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