p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.13C8, C8.27M4(2), C4.13M5(2), (C2×C4)⋊4C16, (C4×C16)⋊3C2, C4○2(C4⋊C16), C8○2(C4⋊C16), C4⋊C16⋊19C2, C4.9(C2×C16), (C4×C8).23C4, C42○(C4⋊C16), C8○2(C22⋊C16), C4○2(C22⋊C16), C8.98(C4○D4), (C22×C4).16C8, C22.5(C2×C16), C2.3(C22×C16), (C22×C8).32C4, C23.38(C2×C8), (C2×C42).50C4, C2.5(C2×M5(2)), C22⋊C16.11C2, C42○(C22⋊C16), C42.341(C2×C4), (C2×C16).66C22, (C4×C8).443C22, (C2×C8).629C23, C4.67(C2×M4(2)), C22.28(C22×C8), C4.76(C42⋊C2), (C22×C8).499C22, C2.4(C42.12C4), (C2×C4×C8).28C2, (C4×C8)○(C4⋊C16), (C4×C8)○(C22⋊C16), (C2×C4).100(C2×C8), (C2×C8).264(C2×C4), (C2×C4).614(C22×C4), (C22×C4).447(C2×C4), SmallGroup(128,894)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.13C8
G = < a,b,c | a4=b4=1, c8=a2, ab=ba, cac-1=a-1b2, bc=cb >
Subgroups: 100 in 80 conjugacy classes, 60 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C22×C4, C4×C8, C2×C16, C2×C42, C22×C8, C4×C16, C22⋊C16, C4⋊C16, C2×C4×C8, C42.13C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C16, C2×C8, M4(2), C22×C4, C4○D4, C2×C16, M5(2), C42⋊C2, C22×C8, C2×M4(2), C42.12C4, C22×C16, C2×M5(2), C42.13C8
►On 64 points
(1 5 9 13)(2 61 10 53)(3 7 11 15)(4 63 12 55)(6 49 14 57)(8 51 16 59)(17 39 25 47)(18 22 26 30)(19 41 27 33)(20 24 28 32)(21 43 29 35)(23 45 31 37)(34 38 42 46)(36 40 44 48)(50 54 58 62)(52 56 60 64)
(1 48 64 22)(2 33 49 23)(3 34 50 24)(4 35 51 25)(5 36 52 26)(6 37 53 27)(7 38 54 28)(8 39 55 29)(9 40 56 30)(10 41 57 31)(11 42 58 32)(12 43 59 17)(13 44 60 18)(14 45 61 19)(15 46 62 20)(16 47 63 21)
(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)| (1,5,9,13)(2,61,10,53)(3,7,11,15)(4,63,12,55)(6,49,14,57)(8,51,16,59)(17,39,25,47)(18,22,26,30)(19,41,27,33)(20,24,28,32)(21,43,29,35)(23,45,31,37)(34,38,42,46)(36,40,44,48)(50,54,58,62)(52,56,60,64), (1,48,64,22)(2,33,49,23)(3,34,50,24)(4,35,51,25)(5,36,52,26)(6,37,53,27)(7,38,54,28)(8,39,55,29)(9,40,56,30)(10,41,57,31)(11,42,58,32)(12,43,59,17)(13,44,60,18)(14,45,61,19)(15,46,62,20)(16,47,63,21), (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( (1,5,9,13)(2,61,10,53)(3,7,11,15)(4,63,12,55)(6,49,14,57)(8,51,16,59)(17,39,25,47)(18,22,26,30)(19,41,27,33)(20,24,28,32)(21,43,29,35)(23,45,31,37)(34,38,42,46)(36,40,44,48)(50,54,58,62)(52,56,60,64), (1,48,64,22)(2,33,49,23)(3,34,50,24)(4,35,51,25)(5,36,52,26)(6,37,53,27)(7,38,54,28)(8,39,55,29)(9,40,56,30)(10,41,57,31)(11,42,58,32)(12,43,59,17)(13,44,60,18)(14,45,61,19)(15,46,62,20)(16,47,63,21), (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([[(1,5,9,13),(2,61,10,53),(3,7,11,15),(4,63,12,55),(6,49,14,57),(8,51,16,59),(17,39,25,47),(18,22,26,30),(19,41,27,33),(20,24,28,32),(21,43,29,35),(23,45,31,37),(34,38,42,46),(36,40,44,48),(50,54,58,62),(52,56,60,64)], [(1,48,64,22),(2,33,49,23),(3,34,50,24),(4,35,51,25),(5,36,52,26),(6,37,53,27),(7,38,54,28),(8,39,55,29),(9,40,56,30),(10,41,57,31),(11,42,58,32),(12,43,59,17),(13,44,60,18),(14,45,61,19),(15,46,62,20),(16,47,63,21)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8X | 16A | ··· | 16AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C16 | M4(2) | C4○D4 | M5(2) |
| kernel | C42.13C8 | C4×C16 | C22⋊C16 | C4⋊C16 | C2×C4×C8 | C4×C8 | C2×C42 | C22×C8 | C42 | C22×C4 | C2×C4 | C8 | C8 | C4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 8 | 8 | 32 | 4 | 4 | 8 |
Matrix representation of C42.13C8 ►in GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 16 | 13 |
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 10 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 1 | 2 |
G:=sub<GL(3,GF(17))| [13,0,0,0,16,0,0,13,1],[4,0,0,0,4,0,0,0,4],[10,0,0,0,15,1,0,0,2] >;
C42.13C8 in GAP, Magma, Sage, TeX
C_4^2._{13}C_8 % in TeX
G:=Group("C4^2.13C8"); // GroupNames label
G:=SmallGroup(128,894);
// by ID
G=gap.SmallGroup(128,894);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,58,102,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^8=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,b*c=c*b>;
// generators/relations