direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42.6C4, C42.676C23, C23.35M4(2), C4⋊C8⋊81C22, (C23×C4).35C4, (C2×C42).56C4, C8⋊C4⋊52C22, C4○(C42.6C4), (C2×C8).396C23, C42.301(C2×C4), (C2×C4).637C24, C24.129(C2×C4), (C2×C4).84M4(2), C4.53(C2×M4(2)), (C22×C42).33C2, C2.8(C22×M4(2)), C4.81(C42⋊C2), C22⋊C8.225C22, (C22×C8).429C22, C23.224(C22×C4), C22.165(C23×C4), (C23×C4).658C22, C22.27(C2×M4(2)), (C2×C42).1105C22, (C22×C4).1271C23, C22.70(C42⋊C2), (C2×C4⋊C8)⋊44C2, (C2×C8⋊C4)⋊30C2, C4.288(C2×C4○D4), (C2×C22⋊C8).45C2, (C2×C4).953(C4○D4), (C2×C4)○(C42.6C4), (C22×C4).459(C2×C4), (C2×C4).497(C22×C4), C2.37(C2×C42⋊C2), SmallGroup(128,1650)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42.6C4
G = < a,b,c,d | a2=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, dcd-1=b2c >
Subgroups: 332 in 244 conjugacy classes, 156 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C22×C8, C23×C4, C2×C8⋊C4, C2×C22⋊C8, C2×C4⋊C8, C42.6C4, C22×C42, C2×C42.6C4
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, C24, C42⋊C2, C2×M4(2), C23×C4, C2×C4○D4, C42.6C4, C2×C42⋊C2, C22×M4(2), C2×C42.6C4
►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 3 5 7)(2 18 6 22)(4 20 8 24)(9 11 13 15)(10 54 14 50)(12 56 16 52)(17 19 21 23)(25 27 29 31)(26 46 30 42)(28 48 32 44)(33 62 37 58)(34 36 38 40)(35 64 39 60)(41 43 45 47)(49 51 53 55)(57 59 61 63)
(1 57 19 34)(2 62 20 39)(3 59 21 36)(4 64 22 33)(5 61 23 38)(6 58 24 35)(7 63 17 40)(8 60 18 37)(9 41 55 27)(10 46 56 32)(11 43 49 29)(12 48 50 26)(13 45 51 31)(14 42 52 28)(15 47 53 25)(16 44 54 30)
(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), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,11,13,15)(10,54,14,50)(12,56,16,52)(17,19,21,23)(25,27,29,31)(26,46,30,42)(28,48,32,44)(33,62,37,58)(34,36,38,40)(35,64,39,60)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,34)(2,62,20,39)(3,59,21,36)(4,64,22,33)(5,61,23,38)(6,58,24,35)(7,63,17,40)(8,60,18,37)(9,41,55,27)(10,46,56,32)(11,43,49,29)(12,48,50,26)(13,45,51,31)(14,42,52,28)(15,47,53,25)(16,44,54,30), (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), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,11,13,15)(10,54,14,50)(12,56,16,52)(17,19,21,23)(25,27,29,31)(26,46,30,42)(28,48,32,44)(33,62,37,58)(34,36,38,40)(35,64,39,60)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,34)(2,62,20,39)(3,59,21,36)(4,64,22,33)(5,61,23,38)(6,58,24,35)(7,63,17,40)(8,60,18,37)(9,41,55,27)(10,46,56,32)(11,43,49,29)(12,48,50,26)(13,45,51,31)(14,42,52,28)(15,47,53,25)(16,44,54,30), (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)], [(1,3,5,7),(2,18,6,22),(4,20,8,24),(9,11,13,15),(10,54,14,50),(12,56,16,52),(17,19,21,23),(25,27,29,31),(26,46,30,42),(28,48,32,44),(33,62,37,58),(34,36,38,40),(35,64,39,60),(41,43,45,47),(49,51,53,55),(57,59,61,63)], [(1,57,19,34),(2,62,20,39),(3,59,21,36),(4,64,22,33),(5,61,23,38),(6,58,24,35),(7,63,17,40),(8,60,18,37),(9,41,55,27),(10,46,56,32),(11,43,49,29),(12,48,50,26),(13,45,51,31),(14,42,52,28),(15,47,53,25),(16,44,54,30)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | M4(2) | C4○D4 | M4(2) |
| kernel | C2×C42.6C4 | C2×C8⋊C4 | C2×C22⋊C8 | C2×C4⋊C8 | C42.6C4 | C22×C42 | C2×C42 | C23×C4 | C2×C4 | C2×C4 | C23 |
| # reps | 1 | 2 | 2 | 2 | 8 | 1 | 12 | 4 | 8 | 8 | 8 |
Matrix representation of C2×C42.6C4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 13 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,13,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,13],[16,0,0,0,0,0,4,0,0,0,0,15,13,0,0,0,0,0,0,13,0,0,0,16,0] >;
C2×C42.6C4 in GAP, Magma, Sage, TeX
C_2\times C_4^2._6C_4
% in TeX
G:=Group("C2xC4^2.6C4"); // GroupNames label
G:=SmallGroup(128,1650);
// by ID
G=gap.SmallGroup(128,1650);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^2,d*c*d^-1=b^2*c>;
// generators/relations