p-group, metabelian, nilpotent (class 2), monomial
Aliases: (C2×D4).5C8, (C2×Q8).5C8, (C22×C16)⋊2C2, C8.132(C2×D4), (C2×C8).388D4, C23.9(C2×C8), C22⋊C16⋊14C2, C2.4(D4○C16), (C2×M5(2))⋊7C2, C8.29(C22⋊C4), C4.14(C22⋊C8), (C2×C8).625C23, (C2×C16).61C22, C4.63(C2×M4(2)), (C2×C4).49M4(2), (C2×M4(2)).28C4, C22.4(C22⋊C8), C22.47(C22×C8), (C22×C8).414C22, (C2×C4).24(C2×C8), (C2×C8).193(C2×C4), (C2×C8○D4).16C2, (C2×C4○D4).17C4, C2.23(C2×C22⋊C8), C4.114(C2×C22⋊C4), (C22×C4).286(C2×C4), (C2×C4).610(C22×C4), (C2×C4).266(C22⋊C4), SmallGroup(128,845)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×D4).5C8
G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=ab-1, dcd-1=ab2c >
Subgroups: 172 in 114 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2×C16, C2×C16, M5(2), C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C22⋊C16, C22×C16, C2×M5(2), C2×C8○D4, (C2×D4).5C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, D4○C16, (C2×D4).5C8
►On 64 points
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 33)(14 34)(15 35)(16 36)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)
(1 13 9 5)(2 42 10 34)(3 15 11 7)(4 44 12 36)(6 46 14 38)(8 48 16 40)(17 52 25 60)(18 22 26 30)(19 54 27 62)(20 24 28 32)(21 56 29 64)(23 58 31 50)(33 45 41 37)(35 47 43 39)(49 53 57 61)(51 55 59 63)
(1 20)(2 52)(3 22)(4 54)(5 24)(6 56)(7 26)(8 58)(9 28)(10 60)(11 30)(12 62)(13 32)(14 64)(15 18)(16 50)(17 42)(19 44)(21 46)(23 48)(25 34)(27 36)(29 38)(31 40)(33 55)(35 57)(37 59)(39 61)(41 63)(43 49)(45 51)(47 53)
(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)
G:=sub<Sym(64)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,33)(14,34)(15,35)(16,36)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55), (1,13,9,5)(2,42,10,34)(3,15,11,7)(4,44,12,36)(6,46,14,38)(8,48,16,40)(17,52,25,60)(18,22,26,30)(19,54,27,62)(20,24,28,32)(21,56,29,64)(23,58,31,50)(33,45,41,37)(35,47,43,39)(49,53,57,61)(51,55,59,63), (1,20)(2,52)(3,22)(4,54)(5,24)(6,56)(7,26)(8,58)(9,28)(10,60)(11,30)(12,62)(13,32)(14,64)(15,18)(16,50)(17,42)(19,44)(21,46)(23,48)(25,34)(27,36)(29,38)(31,40)(33,55)(35,57)(37,59)(39,61)(41,63)(43,49)(45,51)(47,53), (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)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,33)(14,34)(15,35)(16,36)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55), (1,13,9,5)(2,42,10,34)(3,15,11,7)(4,44,12,36)(6,46,14,38)(8,48,16,40)(17,52,25,60)(18,22,26,30)(19,54,27,62)(20,24,28,32)(21,56,29,64)(23,58,31,50)(33,45,41,37)(35,47,43,39)(49,53,57,61)(51,55,59,63), (1,20)(2,52)(3,22)(4,54)(5,24)(6,56)(7,26)(8,58)(9,28)(10,60)(11,30)(12,62)(13,32)(14,64)(15,18)(16,50)(17,42)(19,44)(21,46)(23,48)(25,34)(27,36)(29,38)(31,40)(33,55)(35,57)(37,59)(39,61)(41,63)(43,49)(45,51)(47,53), (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) );
G=PermutationGroup([[(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,33),(14,34),(15,35),(16,36),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55)], [(1,13,9,5),(2,42,10,34),(3,15,11,7),(4,44,12,36),(6,46,14,38),(8,48,16,40),(17,52,25,60),(18,22,26,30),(19,54,27,62),(20,24,28,32),(21,56,29,64),(23,58,31,50),(33,45,41,37),(35,47,43,39),(49,53,57,61),(51,55,59,63)], [(1,20),(2,52),(3,22),(4,54),(5,24),(6,56),(7,26),(8,58),(9,28),(10,60),(11,30),(12,62),(13,32),(14,64),(15,18),(16,50),(17,42),(19,44),(21,46),(23,48),(25,34),(27,36),(29,38),(31,40),(33,55),(35,57),(37,59),(39,61),(41,63),(43,49),(45,51),(47,53)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | M4(2) | D4○C16 |
| kernel | (C2×D4).5C8 | C22⋊C16 | C22×C16 | C2×M5(2) | C2×C8○D4 | C2×M4(2) | C2×C4○D4 | C2×D4 | C2×Q8 | C2×C8 | C2×C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 6 | 2 | 12 | 4 | 4 | 4 | 16 |
Matrix representation of (C2×D4).5C8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 14 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,13,0,0,0,0,4,0,0,0,0,13],[0,4,0,0,13,0,0,0,0,0,0,13,0,0,4,0],[12,0,0,0,0,12,0,0,0,0,0,14,0,0,14,0] >;
(C2×D4).5C8 in GAP, Magma, Sage, TeX
(C_2\times D_4)._5C_8
% in TeX
G:=Group("(C2xD4).5C8"); // GroupNames label
G:=SmallGroup(128,845);
// by ID
G=gap.SmallGroup(128,845);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=a*b^2*c>;
// generators/relations