direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×D4⋊6D4, C22.53C25, C42.552C23, C24.491C23, C23.124C24, C22.792- 1+4, (C2×D4)⋊56D4, D4⋊11(C2×D4), C4⋊Q8⋊81C22, (C2×C4).55C24, C2.20(D4×C23), (C4×D4)⋊102C22, C4⋊C4.465C23, C4⋊D4⋊70C22, C23.484(C2×D4), C4.109(C22×D4), C22⋊Q8⋊83C22, C22.7(C22×D4), (C2×D4).450C23, C22⋊C4.12C23, (C2×Q8).429C23, (C2×C42).924C22, (C23×C4).593C22, C2.11(C2×2- 1+4), (C22×C4).1190C23, (C22×D4).614C22, C22.D4⋊41C22, (C22×Q8).491C22, C4⋊C4○3(C2×D4), D4○2(C2×C4⋊C4), (C2×C4×D4)⋊80C2, C4⋊2(C2×C4○D4), (C2×C4⋊Q8)⋊50C2, C4⋊C4○2(C22×D4), (C2×C4)⋊17(C4○D4), (C2×C4⋊D4)⋊60C2, (C22×C4⋊C4)⋊43C2, (C2×C22⋊Q8)⋊68C2, (C2×C4⋊C4)⋊133C22, (C2×C4).1109(C2×D4), (C22×C4○D4)⋊17C2, (C2×C4○D4)⋊71C22, C2.26(C22×C4○D4), C22.157(C2×C4○D4), (C2×C22.D4)⋊55C2, (C2×C22⋊C4).376C22, SmallGroup(128,2196)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4⋊6D4
G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 1212 in 816 conjugacy classes, 444 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C22×C4⋊C4, C2×C4×D4, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4, C2×C4⋊Q8, D4⋊6D4, C22×C4○D4, C2×D4⋊6D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, C25, D4⋊6D4, D4×C23, C22×C4○D4, C2×2- 1+4, C2×D4⋊6D4
►On 64 points
(1 15)(2 16)(3 13)(4 14)(5 29)(6 30)(7 31)(8 32)(9 35)(10 36)(11 33)(12 34)(17 42)(18 43)(19 44)(20 41)(21 27)(22 28)(23 25)(24 26)(37 45)(38 46)(39 47)(40 48)(49 62)(50 63)(51 64)(52 61)(53 57)(54 58)(55 59)(56 60)
(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 34)(2 33)(3 36)(4 35)(5 28)(6 27)(7 26)(8 25)(9 14)(10 13)(11 16)(12 15)(17 39)(18 38)(19 37)(20 40)(21 30)(22 29)(23 32)(24 31)(41 48)(42 47)(43 46)(44 45)(49 59)(50 58)(51 57)(52 60)(53 64)(54 63)(55 62)(56 61)
(1 51 25 19)(2 52 26 20)(3 49 27 17)(4 50 28 18)(5 38 35 58)(6 39 36 59)(7 40 33 60)(8 37 34 57)(9 54 29 46)(10 55 30 47)(11 56 31 48)(12 53 32 45)(13 62 21 42)(14 63 22 43)(15 64 23 44)(16 61 24 41)
(1 62)(2 63)(3 64)(4 61)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 51)(14 52)(15 49)(16 50)(17 23)(18 24)(19 21)(20 22)(25 42)(26 43)(27 44)(28 41)(29 38)(30 39)(31 40)(32 37)(33 56)(34 53)(35 54)(36 55)
G:=sub<Sym(64)| (1,15)(2,16)(3,13)(4,14)(5,29)(6,30)(7,31)(8,32)(9,35)(10,36)(11,33)(12,34)(17,42)(18,43)(19,44)(20,41)(21,27)(22,28)(23,25)(24,26)(37,45)(38,46)(39,47)(40,48)(49,62)(50,63)(51,64)(52,61)(53,57)(54,58)(55,59)(56,60), (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), (1,34)(2,33)(3,36)(4,35)(5,28)(6,27)(7,26)(8,25)(9,14)(10,13)(11,16)(12,15)(17,39)(18,38)(19,37)(20,40)(21,30)(22,29)(23,32)(24,31)(41,48)(42,47)(43,46)(44,45)(49,59)(50,58)(51,57)(52,60)(53,64)(54,63)(55,62)(56,61), (1,51,25,19)(2,52,26,20)(3,49,27,17)(4,50,28,18)(5,38,35,58)(6,39,36,59)(7,40,33,60)(8,37,34,57)(9,54,29,46)(10,55,30,47)(11,56,31,48)(12,53,32,45)(13,62,21,42)(14,63,22,43)(15,64,23,44)(16,61,24,41), (1,62)(2,63)(3,64)(4,61)(5,46)(6,47)(7,48)(8,45)(9,58)(10,59)(11,60)(12,57)(13,51)(14,52)(15,49)(16,50)(17,23)(18,24)(19,21)(20,22)(25,42)(26,43)(27,44)(28,41)(29,38)(30,39)(31,40)(32,37)(33,56)(34,53)(35,54)(36,55)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,29)(6,30)(7,31)(8,32)(9,35)(10,36)(11,33)(12,34)(17,42)(18,43)(19,44)(20,41)(21,27)(22,28)(23,25)(24,26)(37,45)(38,46)(39,47)(40,48)(49,62)(50,63)(51,64)(52,61)(53,57)(54,58)(55,59)(56,60), (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), (1,34)(2,33)(3,36)(4,35)(5,28)(6,27)(7,26)(8,25)(9,14)(10,13)(11,16)(12,15)(17,39)(18,38)(19,37)(20,40)(21,30)(22,29)(23,32)(24,31)(41,48)(42,47)(43,46)(44,45)(49,59)(50,58)(51,57)(52,60)(53,64)(54,63)(55,62)(56,61), (1,51,25,19)(2,52,26,20)(3,49,27,17)(4,50,28,18)(5,38,35,58)(6,39,36,59)(7,40,33,60)(8,37,34,57)(9,54,29,46)(10,55,30,47)(11,56,31,48)(12,53,32,45)(13,62,21,42)(14,63,22,43)(15,64,23,44)(16,61,24,41), (1,62)(2,63)(3,64)(4,61)(5,46)(6,47)(7,48)(8,45)(9,58)(10,59)(11,60)(12,57)(13,51)(14,52)(15,49)(16,50)(17,23)(18,24)(19,21)(20,22)(25,42)(26,43)(27,44)(28,41)(29,38)(30,39)(31,40)(32,37)(33,56)(34,53)(35,54)(36,55) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,29),(6,30),(7,31),(8,32),(9,35),(10,36),(11,33),(12,34),(17,42),(18,43),(19,44),(20,41),(21,27),(22,28),(23,25),(24,26),(37,45),(38,46),(39,47),(40,48),(49,62),(50,63),(51,64),(52,61),(53,57),(54,58),(55,59),(56,60)], [(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)], [(1,34),(2,33),(3,36),(4,35),(5,28),(6,27),(7,26),(8,25),(9,14),(10,13),(11,16),(12,15),(17,39),(18,38),(19,37),(20,40),(21,30),(22,29),(23,32),(24,31),(41,48),(42,47),(43,46),(44,45),(49,59),(50,58),(51,57),(52,60),(53,64),(54,63),(55,62),(56,61)], [(1,51,25,19),(2,52,26,20),(3,49,27,17),(4,50,28,18),(5,38,35,58),(6,39,36,59),(7,40,33,60),(8,37,34,57),(9,54,29,46),(10,55,30,47),(11,56,31,48),(12,53,32,45),(13,62,21,42),(14,63,22,43),(15,64,23,44),(16,61,24,41)], [(1,62),(2,63),(3,64),(4,61),(5,46),(6,47),(7,48),(8,45),(9,58),(10,59),(11,60),(12,57),(13,51),(14,52),(15,49),(16,50),(17,23),(18,24),(19,21),(20,22),(25,42),(26,43),(27,44),(28,41),(29,38),(30,39),(31,40),(32,37),(33,56),(34,53),(35,54),(36,55)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 2P | 2Q | 2R | 2S | 4A | ··· | 4P | 4Q | ··· | 4AD |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2- 1+4 |
| kernel | C2×D4⋊6D4 | C22×C4⋊C4 | C2×C4×D4 | C2×C4⋊D4 | C2×C22⋊Q8 | C2×C22.D4 | C2×C4⋊Q8 | D4⋊6D4 | C22×C4○D4 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 16 | 2 | 8 | 8 | 2 |
Matrix representation of C2×D4⋊6D4 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 1 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,2,3],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,1,0,0,0,2,3],[4,0,0,0,0,0,2,0,0,0,0,2,3,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,3,4,0,0,0,3,2,0,0,0,0,0,4,0,0,0,0,4,1] >;
C2×D4⋊6D4 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_6D_4
% in TeX
G:=Group("C2xD4:6D4"); // GroupNames label
G:=SmallGroup(128,2196);
// by ID
G=gap.SmallGroup(128,2196);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,184,570]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations