direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42.6C22, C42.256C23, C4⋊C8⋊80C22, C24.98(C2×C4), (C23×C8).12C2, C4.59(C22×Q8), C23.74(C4⋊C4), (C2×C4).633C24, (C2×C8).394C23, C42.200(C2×C4), (C22×C4).783D4, C4.185(C22×D4), (C22×C4).100Q8, C42⋊C2.26C4, C22.38(C8○D4), C4○(C42.6C22), (C2×C42).751C22, (C22×C8).505C22, C23.291(C22×C4), C22.162(C23×C4), (C23×C4).693C22, (C22×C4).1501C23, (C22×M4(2)).28C2, C42⋊C2.282C22, (C2×M4(2)).336C22, (C2×C4⋊C8)⋊42C2, C4.61(C2×C4⋊C4), C2.8(C2×C8○D4), (C2×C4⋊C4).65C4, C4⋊C4.212(C2×C4), C22.33(C2×C4⋊C4), C2.19(C22×C4⋊C4), (C2×C4).235(C2×Q8), (C2×C4).148(C4⋊C4), (C2×C4).1408(C2×D4), (C2×C22⋊C4).42C4, C22⋊C4.63(C2×C4), (C2×C4).247(C22×C4), (C22×C4).327(C2×C4), (C2×C42⋊C2).51C2, (C2×C4)○(C42.6C22), SmallGroup(128,1636)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42.6C22
G = < a,b,c,d,e | a2=b4=c4=1, d2=c, e2=b2c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe-1=bc2, cd=dc, ce=ec, ede-1=b2c2d >
Subgroups: 332 in 256 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C2×C4⋊C8, C42.6C22, C2×C42⋊C2, C23×C8, C22×M4(2), C2×C42.6C22
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C8○D4, C23×C4, C22×D4, C22×Q8, C42.6C22, C22×C4⋊C4, C2×C8○D4, C2×C42.6C22
►On 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 59 19 39)(2 36 20 64)(3 61 21 33)(4 38 22 58)(5 63 23 35)(6 40 24 60)(7 57 17 37)(8 34 18 62)(9 27 43 55)(10 52 44 32)(11 29 45 49)(12 54 46 26)(13 31 47 51)(14 56 48 28)(15 25 41 53)(16 50 42 30)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 63 23 39)(2 40 24 64)(3 57 17 33)(4 34 18 58)(5 59 19 35)(6 36 20 60)(7 61 21 37)(8 38 22 62)(9 27 47 51)(10 52 48 28)(11 29 41 53)(12 54 42 30)(13 31 43 55)(14 56 44 32)(15 25 45 49)(16 50 46 26)
G:=sub<Sym(64)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,59,19,39)(2,36,20,64)(3,61,21,33)(4,38,22,58)(5,63,23,35)(6,40,24,60)(7,57,17,37)(8,34,18,62)(9,27,43,55)(10,52,44,32)(11,29,45,49)(12,54,46,26)(13,31,47,51)(14,56,48,28)(15,25,41,53)(16,50,42,30), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,63,23,39)(2,40,24,64)(3,57,17,33)(4,34,18,58)(5,59,19,35)(6,36,20,60)(7,61,21,37)(8,38,22,62)(9,27,47,51)(10,52,48,28)(11,29,41,53)(12,54,42,30)(13,31,43,55)(14,56,44,32)(15,25,45,49)(16,50,46,26)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,59,19,39)(2,36,20,64)(3,61,21,33)(4,38,22,58)(5,63,23,35)(6,40,24,60)(7,57,17,37)(8,34,18,62)(9,27,43,55)(10,52,44,32)(11,29,45,49)(12,54,46,26)(13,31,47,51)(14,56,48,28)(15,25,41,53)(16,50,42,30), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,63,23,39)(2,40,24,64)(3,57,17,33)(4,34,18,58)(5,59,19,35)(6,36,20,60)(7,61,21,37)(8,38,22,62)(9,27,47,51)(10,52,48,28)(11,29,41,53)(12,54,42,30)(13,31,43,55)(14,56,44,32)(15,25,45,49)(16,50,46,26) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,59,19,39),(2,36,20,64),(3,61,21,33),(4,38,22,58),(5,63,23,35),(6,40,24,60),(7,57,17,37),(8,34,18,62),(9,27,43,55),(10,52,44,32),(11,29,45,49),(12,54,46,26),(13,31,47,51),(14,56,48,28),(15,25,41,53),(16,50,42,30)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,63,23,39),(2,40,24,64),(3,57,17,33),(4,34,18,58),(5,59,19,35),(6,36,20,60),(7,61,21,37),(8,38,22,62),(9,27,47,51),(10,52,48,28),(11,29,41,53),(12,54,42,30),(13,31,43,55),(14,56,44,32),(15,25,45,49),(16,50,46,26)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 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 | C2 | C4 | C4 | C4 | D4 | Q8 | C8○D4 |
| kernel | C2×C42.6C22 | C2×C4⋊C8 | C42.6C22 | C2×C42⋊C2 | C23×C8 | C22×M4(2) | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C22×C4 | C22×C4 | C22 |
| # reps | 1 | 4 | 8 | 1 | 1 | 1 | 4 | 4 | 8 | 4 | 4 | 16 |
Matrix representation of C2×C42.6C22 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,2,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,15],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
C2×C42.6C22 in GAP, Magma, Sage, TeX
C_2\times C_4^2._6C_2^2
% in TeX
G:=Group("C2xC4^2.6C2^2"); // GroupNames label
G:=SmallGroup(128,1636);
// by ID
G=gap.SmallGroup(128,1636);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=1,d^2=c,e^2=b^2*c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1*c^2,e*b*e^-1=b*c^2,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*c^2*d>;
// generators/relations