p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8.12M4(2), (C4×C16)⋊4C2, C4⋊C4.8C8, C4⋊C16⋊16C2, C16⋊5C4⋊9C2, C22⋊C4.4C8, C8⋊C4.18C4, C2.6(D4○C16), C23.10(C2×C8), C8.100(C4○D4), C22⋊C16.10C2, (C2×C16).67C22, (C4×C8).322C22, C42.245(C2×C4), (C2×C8).631C23, C4.69(C2×M4(2)), C42⋊C2.23C4, (C2×M4(2)).34C4, C22.51(C22×C8), C4.78(C42⋊C2), C8○2M4(2).21C2, (C22×C8).419C22, C2.12(C42.12C4), (C2×C4).28(C2×C8), (C2×C8).196(C2×C4), (C22×C4).289(C2×C4), (C2×C4).616(C22×C4), SmallGroup(128,896)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.12M4(2)
G = < a,b,c | a8=c2=1, b8=a4, bab-1=cac=a5, cbc=a6b5 >
Subgroups: 92 in 69 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, C42⋊C2, C22×C8, C2×M4(2), C4×C16, C16⋊5C4, C22⋊C16, C4⋊C16, C8○2M4(2), C8.12M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, C42⋊C2, C22×C8, C2×M4(2), C42.12C4, D4○C16, C8.12M4(2)
►On 64 points
(1 18 54 46 9 26 62 38)(2 27 55 39 10 19 63 47)(3 20 56 48 11 28 64 40)(4 29 57 41 12 21 49 33)(5 22 58 34 13 30 50 42)(6 31 59 43 14 23 51 35)(7 24 60 36 15 32 52 44)(8 17 61 45 16 25 53 37)
(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)
(2 51)(4 53)(6 55)(8 57)(10 59)(12 61)(14 63)(16 49)(17 33)(18 26)(19 35)(20 28)(21 37)(22 30)(23 39)(24 32)(25 41)(27 43)(29 45)(31 47)(34 42)(36 44)(38 46)(40 48)
G:=sub<Sym(64)| (1,18,54,46,9,26,62,38)(2,27,55,39,10,19,63,47)(3,20,56,48,11,28,64,40)(4,29,57,41,12,21,49,33)(5,22,58,34,13,30,50,42)(6,31,59,43,14,23,51,35)(7,24,60,36,15,32,52,44)(8,17,61,45,16,25,53,37), (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), (2,51)(4,53)(6,55)(8,57)(10,59)(12,61)(14,63)(16,49)(17,33)(18,26)(19,35)(20,28)(21,37)(22,30)(23,39)(24,32)(25,41)(27,43)(29,45)(31,47)(34,42)(36,44)(38,46)(40,48)>;
G:=Group( (1,18,54,46,9,26,62,38)(2,27,55,39,10,19,63,47)(3,20,56,48,11,28,64,40)(4,29,57,41,12,21,49,33)(5,22,58,34,13,30,50,42)(6,31,59,43,14,23,51,35)(7,24,60,36,15,32,52,44)(8,17,61,45,16,25,53,37), (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), (2,51)(4,53)(6,55)(8,57)(10,59)(12,61)(14,63)(16,49)(17,33)(18,26)(19,35)(20,28)(21,37)(22,30)(23,39)(24,32)(25,41)(27,43)(29,45)(31,47)(34,42)(36,44)(38,46)(40,48) );
G=PermutationGroup([[(1,18,54,46,9,26,62,38),(2,27,55,39,10,19,63,47),(3,20,56,48,11,28,64,40),(4,29,57,41,12,21,49,33),(5,22,58,34,13,30,50,42),(6,31,59,43,14,23,51,35),(7,24,60,36,15,32,52,44),(8,17,61,45,16,25,53,37)], [(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)], [(2,51),(4,53),(6,55),(8,57),(10,59),(12,61),(14,63),(16,49),(17,33),(18,26),(19,35),(20,28),(21,37),(22,30),(23,39),(24,32),(25,41),(27,43),(29,45),(31,47),(34,42),(36,44),(38,46),(40,48)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 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 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | M4(2) | C4○D4 | D4○C16 |
| kernel | C8.12M4(2) | C4×C16 | C16⋊5C4 | C22⋊C16 | C4⋊C16 | C8○2M4(2) | C8⋊C4 | C42⋊C2 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C8 | C8 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 8 | 8 | 4 | 4 | 16 |
Matrix representation of C8.12M4(2) ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 3 |
| 0 | 0 | 6 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,2,16,0,0,0,15],[0,1,0,0,1,0,0,0,0,0,5,6,0,0,3,12],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,4,16] >;
C8.12M4(2) in GAP, Magma, Sage, TeX
C_8._{12}M_4(2) % in TeX
G:=Group("C8.12M4(2)"); // GroupNames label
G:=SmallGroup(128,896);
// by ID
G=gap.SmallGroup(128,896);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,58,102,124]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,b*a*b^-1=c*a*c=a^5,c*b*c=a^6*b^5>;
// generators/relations