p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.D4⋊3C2, (C2×C8).144D4, C8.110(C2×D4), (C2×D4).218D4, C4⋊C4.28C23, (C2×Q8).173D4, C2.10(Q8○D8), C23.77(C2×D4), C8.18D4⋊34C2, C4.71(C4⋊D4), (C2×C4).263C24, (C2×C8).255C23, (C22×Q16)⋊16C2, C4.157(C22×D4), (C2×Q8).53C23, C4.Q8.130C22, C2.D8.162C22, C22⋊Q8.19C22, C23.38D4⋊35C2, C23.25D4⋊29C2, C22.88(C4⋊D4), (C22×C8).260C22, (C22×C4).985C23, (C2×Q16).117C22, C22.523(C22×D4), Q8⋊C4.122C22, (C22×Q8).283C22, C42⋊C2.112C22, (C2×M4(2)).265C22, C23.38C23.11C2, (C2×C8○D4).9C2, C4.30(C2×C4○D4), (C2×C4).131(C2×D4), C2.81(C2×C4⋊D4), (C2×C4).284(C4○D4), (C2×C4○D4).303C22, SmallGroup(128,1791)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 396 in 230 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×14], C23, C23 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×4], Q8⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22⋊Q8 [×8], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×Q16 [×4], C2×Q16 [×4], C22×Q8 [×2], C2×C4○D4, C23.38D4 [×2], C23.25D4, C8.18D4 [×4], C8.D4 [×4], C23.38C23 [×2], C2×C8○D4, C22×Q16, C8.D4⋊C2
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, Q8○D8 [×2], C8.D4⋊C2
Generators and relations
G = < a,b,c,d | a8=b4=d2=1, c2=a4, bab-1=a3, cac-1=a-1, ad=da, cbc-1=a4b-1, dbd=a4b, cd=dc >
►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 57 43 36)(2 60 44 39)(3 63 45 34)(4 58 46 37)(5 61 47 40)(6 64 48 35)(7 59 41 38)(8 62 42 33)(9 25 20 49)(10 28 21 52)(11 31 22 55)(12 26 23 50)(13 29 24 53)(14 32 17 56)(15 27 18 51)(16 30 19 54)
(1 40 5 36)(2 39 6 35)(3 38 7 34)(4 37 8 33)(9 53 13 49)(10 52 14 56)(11 51 15 55)(12 50 16 54)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(41 63 45 59)(42 62 46 58)(43 61 47 57)(44 60 48 64)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
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), (1,57,43,36)(2,60,44,39)(3,63,45,34)(4,58,46,37)(5,61,47,40)(6,64,48,35)(7,59,41,38)(8,62,42,33)(9,25,20,49)(10,28,21,52)(11,31,22,55)(12,26,23,50)(13,29,24,53)(14,32,17,56)(15,27,18,51)(16,30,19,54), (1,40,5,36)(2,39,6,35)(3,38,7,34)(4,37,8,33)(9,53,13,49)(10,52,14,56)(11,51,15,55)(12,50,16,54)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(41,63,45,59)(42,62,46,58)(43,61,47,57)(44,60,48,64), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50)>;
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), (1,57,43,36)(2,60,44,39)(3,63,45,34)(4,58,46,37)(5,61,47,40)(6,64,48,35)(7,59,41,38)(8,62,42,33)(9,25,20,49)(10,28,21,52)(11,31,22,55)(12,26,23,50)(13,29,24,53)(14,32,17,56)(15,27,18,51)(16,30,19,54), (1,40,5,36)(2,39,6,35)(3,38,7,34)(4,37,8,33)(9,53,13,49)(10,52,14,56)(11,51,15,55)(12,50,16,54)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(41,63,45,59)(42,62,46,58)(43,61,47,57)(44,60,48,64), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50) );
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)], [(1,57,43,36),(2,60,44,39),(3,63,45,34),(4,58,46,37),(5,61,47,40),(6,64,48,35),(7,59,41,38),(8,62,42,33),(9,25,20,49),(10,28,21,52),(11,31,22,55),(12,26,23,50),(13,29,24,53),(14,32,17,56),(15,27,18,51),(16,30,19,54)], [(1,40,5,36),(2,39,6,35),(3,38,7,34),(4,37,8,33),(9,53,13,49),(10,52,14,56),(11,51,15,55),(12,50,16,54),(17,32,21,28),(18,31,22,27),(19,30,23,26),(20,29,24,25),(41,63,45,59),(42,62,46,58),(43,61,47,57),(44,60,48,64)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,45),(18,46),(19,47),(20,48),(21,41),(22,42),(23,43),(24,44),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 6 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 16 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 0 | 1 | 10 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 15 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 16 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 16 |
| 0 | 0 | 0 | 0 | 16 | 7 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,3,3,0,0,0,0,14,3,0,0],[3,6,0,0,0,0,4,14,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,1,10],[3,15,0,0,0,0,4,14,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | Q8○D8 |
| kernel | C8.D4⋊C2 | C23.38D4 | C23.25D4 | C8.18D4 | C8.D4 | C23.38C23 | C2×C8○D4 | C22×Q16 | C2×C8 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_8.D_4\rtimes C_2
% in TeX
G:=Group("C8.D4:C2"); // GroupNames label
G:=SmallGroup(128,1791);
// by ID
G=gap.SmallGroup(128,1791);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,521,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^4=d^2=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^4*b^-1,d*b*d=a^4*b,c*d=d*c>;
// generators/relations