p-group, metabelian, nilpotent (class 2), monomial
Aliases: (C2×C8).195D4, (C2×C4)⋊2M4(2), C24.55(C2×C4), C4.115C22≀C2, (C22×C4).47Q8, C2.16(C8⋊9D4), C2.12(C8⋊6D4), C23.28(C4⋊C4), (C22×C4).281D4, C22.143(C4×D4), C22.51(C8○D4), C4.112(C22⋊Q8), (C22×C8).26C22, C2.C42.20C4, C23.312(C22×C4), (C2×C42).262C22, (C23×C4).247C22, C22.65(C2×M4(2)), C2.7(C23.8Q8), C2.10(C4⋊M4(2)), C22.7C42⋊39C2, (C22×C4).1628C23, (C22×M4(2)).19C2, C4.132(C22.D4), C2.13(C42.6C22), (C2×C4⋊C8)⋊39C2, (C2×C4).50(C4⋊C4), (C2×C4).342(C2×Q8), (C2×C4).1526(C2×D4), (C4×C22⋊C4).14C2, (C2×C22⋊C8).21C2, (C2×C22⋊C4).38C4, C22.108(C2×C4⋊C4), (C2×C4).934(C4○D4), (C22×C4).117(C2×C4), SmallGroup(128,583)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).195D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=b2, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=ab5, dcd-1=b6c3 >
Subgroups: 276 in 162 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C22×C8, C22×C8, C2×M4(2), C23×C4, C22.7C42, C4×C22⋊C4, C2×C22⋊C8, C2×C4⋊C8, C22×M4(2), (C2×C8).195D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×M4(2), C8○D4, C23.8Q8, C4⋊M4(2), C42.6C22, C8⋊9D4, C8⋊6D4, (C2×C8).195D4
►On 64 points
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(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 56 15 19 5 52 11 23)(2 33 16 64 6 37 12 60)(3 50 9 21 7 54 13 17)(4 35 10 58 8 39 14 62)(18 42 51 30 22 46 55 26)(20 44 53 32 24 48 49 28)(25 61 41 34 29 57 45 38)(27 63 43 36 31 59 47 40)
(1 17 3 19 5 21 7 23)(2 58 4 60 6 62 8 64)(9 52 11 54 13 56 15 50)(10 33 12 35 14 37 16 39)(18 46 20 48 22 42 24 44)(25 40 27 34 29 36 31 38)(26 53 28 55 30 49 32 51)(41 63 43 57 45 59 47 61)
G:=sub<Sym(64)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (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), (1,56,15,19,5,52,11,23)(2,33,16,64,6,37,12,60)(3,50,9,21,7,54,13,17)(4,35,10,58,8,39,14,62)(18,42,51,30,22,46,55,26)(20,44,53,32,24,48,49,28)(25,61,41,34,29,57,45,38)(27,63,43,36,31,59,47,40), (1,17,3,19,5,21,7,23)(2,58,4,60,6,62,8,64)(9,52,11,54,13,56,15,50)(10,33,12,35,14,37,16,39)(18,46,20,48,22,42,24,44)(25,40,27,34,29,36,31,38)(26,53,28,55,30,49,32,51)(41,63,43,57,45,59,47,61)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (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), (1,56,15,19,5,52,11,23)(2,33,16,64,6,37,12,60)(3,50,9,21,7,54,13,17)(4,35,10,58,8,39,14,62)(18,42,51,30,22,46,55,26)(20,44,53,32,24,48,49,28)(25,61,41,34,29,57,45,38)(27,63,43,36,31,59,47,40), (1,17,3,19,5,21,7,23)(2,58,4,60,6,62,8,64)(9,52,11,54,13,56,15,50)(10,33,12,35,14,37,16,39)(18,46,20,48,22,42,24,44)(25,40,27,34,29,36,31,38)(26,53,28,55,30,49,32,51)(41,63,43,57,45,59,47,61) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52)], [(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)], [(1,56,15,19,5,52,11,23),(2,33,16,64,6,37,12,60),(3,50,9,21,7,54,13,17),(4,35,10,58,8,39,14,62),(18,42,51,30,22,46,55,26),(20,44,53,32,24,48,49,28),(25,61,41,34,29,57,45,38),(27,63,43,36,31,59,47,40)], [(1,17,3,19,5,21,7,23),(2,58,4,60,6,62,8,64),(9,52,11,54,13,56,15,50),(10,33,12,35,14,37,16,39),(18,46,20,48,22,42,24,44),(25,40,27,34,29,36,31,38),(26,53,28,55,30,49,32,51),(41,63,43,57,45,59,47,61)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | (C2×C8).195D4 | C22.7C42 | C4×C22⋊C4 | C2×C22⋊C8 | C2×C4⋊C8 | C22×M4(2) | C2.C42 | C2×C22⋊C4 | C2×C8 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 8 | 4 | 8 |
Matrix representation of (C2×C8).195D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 10 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 10 | 4 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,16,0,0,0,0,0,0,0,1,10,0,0,0,0,2,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,16,0,0,0,0,0,0,0,13,11,0,0,0,0,9,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,16,0,0,0,0,0,0,0,13,10,0,0,0,0,9,4] >;
(C2×C8).195D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{195}D_4 % in TeX
G:=Group("(C2xC8).195D4"); // GroupNames label
G:=SmallGroup(128,583);
// by ID
G=gap.SmallGroup(128,583);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=b^2,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*c*d^-1=b^6*c^3>;
// generators/relations