direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×D4.2D4, C42.207D4, C42.316C23, D4.2(C2×D4), C4⋊C8⋊63C22, (C2×D4).215D4, (C4×D4)⋊82C22, (C22×D8).8C2, C4.62(C22×D4), C4.65(C4⋊D4), C4⋊C4.372C23, (C2×C4).235C24, (C2×C8).304C23, C23.854(C2×D4), (C22×C4).716D4, (C2×Q8).32C23, D4⋊C4⋊73C22, Q8⋊C4⋊67C22, (C22×SD16)⋊21C2, (C2×SD16)⋊72C22, (C2×D8).116C22, (C2×D4).385C23, C22.88(C4○D8), C4.4D4⋊52C22, (C2×C42).804C22, (C22×C8).137C22, C22.495(C22×D4), C22.167(C4⋊D4), C22.113(C8⋊C22), (C22×C4).1525C23, (C22×D4).335C22, (C22×Q8).268C22, (C2×C4×D4)⋊60C2, (C2×C4⋊C8)⋊25C2, C2.10(C2×C4○D8), C4.145(C2×C4○D4), (C2×C4).463(C2×D4), C2.53(C2×C4⋊D4), C2.13(C2×C8⋊C22), (C2×D4⋊C4)⋊37C2, (C2×Q8⋊C4)⋊23C2, (C2×C4.4D4)⋊38C2, (C2×C4).902(C4○D4), (C2×C4⋊C4).916C22, SmallGroup(128,1763)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4.2D4
G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >
Subgroups: 604 in 282 conjugacy classes, 108 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4.4D4, C4.4D4, C22×C8, C2×D8, C2×D8, C2×SD16, C2×SD16, C23×C4, C22×D4, C22×Q8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4⋊C8, D4.2D4, C2×C4×D4, C2×C4.4D4, C22×D8, C22×SD16, C2×D4.2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, D4.2D4, C2×C4⋊D4, C2×C4○D8, C2×C8⋊C22, C2×D4.2D4
►On 64 points
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)(17 31)(18 32)(19 29)(20 30)(21 27)(22 28)(23 25)(24 26)(33 47)(34 48)(35 45)(36 46)(37 43)(38 44)(39 41)(40 42)(49 63)(50 64)(51 61)(52 62)(53 59)(54 60)(55 57)(56 58)
(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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 32)(8 31)(9 18)(10 17)(11 20)(12 19)(13 22)(14 21)(15 24)(16 23)(33 58)(34 57)(35 60)(36 59)(37 62)(38 61)(39 64)(40 63)(41 50)(42 49)(43 52)(44 51)(45 54)(46 53)(47 56)(48 55)
(1 45 5 43)(2 46 6 44)(3 47 7 41)(4 48 8 42)(9 39 15 33)(10 40 16 34)(11 37 13 35)(12 38 14 36)(17 63 23 57)(18 64 24 58)(19 61 21 59)(20 62 22 60)(25 55 31 49)(26 56 32 50)(27 53 29 51)(28 54 30 52)
(1 33 3 35)(2 36 4 34)(5 39 7 37)(6 38 8 40)(9 43 11 41)(10 42 12 44)(13 47 15 45)(14 46 16 48)(17 52 19 50)(18 51 20 49)(21 56 23 54)(22 55 24 53)(25 60 27 58)(26 59 28 57)(29 64 31 62)(30 63 32 61)
G:=sub<Sym(64)| (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10)(17,31)(18,32)(19,29)(20,30)(21,27)(22,28)(23,25)(24,26)(33,47)(34,48)(35,45)(36,46)(37,43)(38,44)(39,41)(40,42)(49,63)(50,64)(51,61)(52,62)(53,59)(54,60)(55,57)(56,58), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,32)(8,31)(9,18)(10,17)(11,20)(12,19)(13,22)(14,21)(15,24)(16,23)(33,58)(34,57)(35,60)(36,59)(37,62)(38,61)(39,64)(40,63)(41,50)(42,49)(43,52)(44,51)(45,54)(46,53)(47,56)(48,55), (1,45,5,43)(2,46,6,44)(3,47,7,41)(4,48,8,42)(9,39,15,33)(10,40,16,34)(11,37,13,35)(12,38,14,36)(17,63,23,57)(18,64,24,58)(19,61,21,59)(20,62,22,60)(25,55,31,49)(26,56,32,50)(27,53,29,51)(28,54,30,52), (1,33,3,35)(2,36,4,34)(5,39,7,37)(6,38,8,40)(9,43,11,41)(10,42,12,44)(13,47,15,45)(14,46,16,48)(17,52,19,50)(18,51,20,49)(21,56,23,54)(22,55,24,53)(25,60,27,58)(26,59,28,57)(29,64,31,62)(30,63,32,61)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10)(17,31)(18,32)(19,29)(20,30)(21,27)(22,28)(23,25)(24,26)(33,47)(34,48)(35,45)(36,46)(37,43)(38,44)(39,41)(40,42)(49,63)(50,64)(51,61)(52,62)(53,59)(54,60)(55,57)(56,58), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,32)(8,31)(9,18)(10,17)(11,20)(12,19)(13,22)(14,21)(15,24)(16,23)(33,58)(34,57)(35,60)(36,59)(37,62)(38,61)(39,64)(40,63)(41,50)(42,49)(43,52)(44,51)(45,54)(46,53)(47,56)(48,55), (1,45,5,43)(2,46,6,44)(3,47,7,41)(4,48,8,42)(9,39,15,33)(10,40,16,34)(11,37,13,35)(12,38,14,36)(17,63,23,57)(18,64,24,58)(19,61,21,59)(20,62,22,60)(25,55,31,49)(26,56,32,50)(27,53,29,51)(28,54,30,52), (1,33,3,35)(2,36,4,34)(5,39,7,37)(6,38,8,40)(9,43,11,41)(10,42,12,44)(13,47,15,45)(14,46,16,48)(17,52,19,50)(18,51,20,49)(21,56,23,54)(22,55,24,53)(25,60,27,58)(26,59,28,57)(29,64,31,62)(30,63,32,61) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,11),(6,12),(7,9),(8,10),(17,31),(18,32),(19,29),(20,30),(21,27),(22,28),(23,25),(24,26),(33,47),(34,48),(35,45),(36,46),(37,43),(38,44),(39,41),(40,42),(49,63),(50,64),(51,61),(52,62),(53,59),(54,60),(55,57),(56,58)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,30),(6,29),(7,32),(8,31),(9,18),(10,17),(11,20),(12,19),(13,22),(14,21),(15,24),(16,23),(33,58),(34,57),(35,60),(36,59),(37,62),(38,61),(39,64),(40,63),(41,50),(42,49),(43,52),(44,51),(45,54),(46,53),(47,56),(48,55)], [(1,45,5,43),(2,46,6,44),(3,47,7,41),(4,48,8,42),(9,39,15,33),(10,40,16,34),(11,37,13,35),(12,38,14,36),(17,63,23,57),(18,64,24,58),(19,61,21,59),(20,62,22,60),(25,55,31,49),(26,56,32,50),(27,53,29,51),(28,54,30,52)], [(1,33,3,35),(2,36,4,34),(5,39,7,37),(6,38,8,40),(9,43,11,41),(10,42,12,44),(13,47,15,45),(14,46,16,48),(17,52,19,50),(18,51,20,49),(21,56,23,54),(22,55,24,53),(25,60,27,58),(26,59,28,57),(29,64,31,62),(30,63,32,61)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C2×D4.2D4 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4⋊C8 | D4.2D4 | C2×C4×D4 | C2×C4.4D4 | C22×D8 | C22×SD16 | C42 | C22×C4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C2×D4.2D4 ►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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 3 | 6 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 | 4 |
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],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,11,3,0,0,0,11,6],[16,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,13],[16,0,0,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,13,4,0,0,0,0,4] >;
C2×D4.2D4 in GAP, Magma, Sage, TeX
C_2\times D_4._2D_4
% in TeX
G:=Group("C2xD4.2D4"); // GroupNames label
G:=SmallGroup(128,1763);
// by ID
G=gap.SmallGroup(128,1763);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^-1>;
// generators/relations