direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4⋊C8, C42.10C4, C42.66C22, C22.10M4(2), C4○(C4⋊C8), (C2×C4)⋊3C8, C4⋊2(C2×C8), C4.71(C2×D4), C4.21(C4⋊C4), C4.20(C2×Q8), (C2×C4).25Q8, (C2×C4).146D4, C2.2(C22×C8), (C22×C8).6C2, C23.39(C2×C4), (C2×C42).13C2, (C22×C4).14C4, C22.10(C2×C8), (C2×C8).60C22, C2.4(C2×M4(2)), C22.18(C4⋊C4), (C2×C4).147C23, C22.21(C22×C4), (C22×C4).136C22, (C2×C4)○(C4⋊C8), C2.3(C2×C4⋊C4), (C2×C4).85(C2×C4), SmallGroup(64,103)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊C8
G = < a,b,c | a2=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 81 in 69 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C4⋊C8, C2×C42, C22×C8, C2×C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8
►Regular action on 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(1 59 23 40)(2 33 24 60)(3 61 17 34)(4 35 18 62)(5 63 19 36)(6 37 20 64)(7 57 21 38)(8 39 22 58)(9 43 51 25)(10 26 52 44)(11 45 53 27)(12 28 54 46)(13 47 55 29)(14 30 56 48)(15 41 49 31)(16 32 50 42)
(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)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,59,23,40)(2,33,24,60)(3,61,17,34)(4,35,18,62)(5,63,19,36)(6,37,20,64)(7,57,21,38)(8,39,22,58)(9,43,51,25)(10,26,52,44)(11,45,53,27)(12,28,54,46)(13,47,55,29)(14,30,56,48)(15,41,49,31)(16,32,50,42), (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( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,59,23,40)(2,33,24,60)(3,61,17,34)(4,35,18,62)(5,63,19,36)(6,37,20,64)(7,57,21,38)(8,39,22,58)(9,43,51,25)(10,26,52,44)(11,45,53,27)(12,28,54,46)(13,47,55,29)(14,30,56,48)(15,41,49,31)(16,32,50,42), (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([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,23),(10,24),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(1,59,23,40),(2,33,24,60),(3,61,17,34),(4,35,18,62),(5,63,19,36),(6,37,20,64),(7,57,21,38),(8,39,22,58),(9,43,51,25),(10,26,52,44),(11,45,53,27),(12,28,54,46),(13,47,55,29),(14,30,56,48),(15,41,49,31),(16,32,50,42)], [(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)]])
C2×C4⋊C8 is a maximal subgroup of
C42.385D4 C42.46Q8 C42.3Q8 C42.25D4 C42.27D4 C42.8Q8 C42.389D4 C22.M5(2) C42.397D4 C42.45D4 C42.46D4 C42.47D4 C42.409D4 C42.410D4 C42.78D4 C42.79D4 C42.80D4 C42.81D4 M4(2)⋊1C8 C42.90D4 C42.91D4 C42.92D4 C43.7C2 C42.45Q8 C8×C4⋊C4 C4⋊C8⋊13C4 C4⋊C8⋊14C4 C42.425D4 C42.95D4 C42.98D4 C42.99D4 C42.100D4 C42.101D4 C42⋊8C8 C42.23Q8 C42⋊9C8 C42.25Q8 C23.21M4(2) (C2×C8).195D4 C4⋊C4⋊3C8 (C2×C8).Q8 C22⋊C4⋊4C8 C23.9M4(2) C42.61Q8 C42.27Q8 C42.29Q8 C42.30Q8 C42.31Q8 C42.430D4 C42.325D4 C42.109D4 C42.117D4 C42.118D4 C42.119D4 C42.327D4 C42.120D4 C42.121D4 C42.122D4 C42.123D4 (C2×C4)⋊3D8 (C2×C4)⋊5SD16 (C2×C4)⋊3Q16 C4⋊C4.106D4 (C2×Q8).8Q8 (C2×C4).23D8 (C2×C8).52D4 (C2×C4).26D8 (C2×C4).21Q16 C4.(C4⋊Q8) C42.674C23 C42.678C23 D4×C2×C8 M4(2)⋊23D4 Q8×C2×C8 M4(2)⋊9Q8 C42.697C23 C42.698C23 D4⋊8M4(2) C42.307C23 C42.309C23 C42.443D4 C42.447D4 C42.293D4 C42.294D4 C42.295D4 C42.296D4 C42.297D4 C42.298D4 C42.11F5
C2×C4⋊C8 is a maximal quotient of
C42.42Q8 M4(2)⋊1C8 C42.425D4 C42⋊8C8 C42⋊9C8 C23.21M4(2) C42.61Q8 C4⋊M5(2) C4⋊C4.7C8 M4(2).1C8 C42.11F5
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | D4 | Q8 | M4(2) |
| kernel | C2×C4⋊C8 | C4⋊C8 | C2×C42 | C22×C8 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of C2×C4⋊C8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[9,0,0,0,0,4,0,0,0,0,15,0,0,0,0,2] >;
C2×C4⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes C_8
% in TeX
G:=Group("C2xC4:C8"); // GroupNames label
G:=SmallGroup(64,103);
// by ID
G=gap.SmallGroup(64,103);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,88]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations