direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×D4.Q8, C42.215D4, C42.328C23, D4.1(C2×Q8), C4⋊C8⋊65C22, (C2×D4).34Q8, C4⋊C4.35C23, C4.Q8⋊58C22, C2.D8⋊49C22, C4.23(C22×Q8), (C2×C8).307C23, (C2×C4).270C24, C23.866(C2×D4), (C22×C4).718D4, C4.63(C22⋊Q8), (C2×D4).392C23, (C4×D4).314C22, C22.93(C4○D8), C42.C2⋊29C22, (C22×C8).142C22, (C2×C42).816C22, C22.530(C22×D4), C22.98(C22⋊Q8), D4⋊C4.179C22, C22.118(C8⋊C22), (C22×C4).1540C23, (C22×D4).570C22, (C2×C4⋊C8)⋊28C2, (C2×C4×D4).82C2, (C2×C4.Q8)⋊29C2, (C2×C2.D8)⋊21C2, C2.17(C2×C4○D8), C4.80(C2×C4○D4), (C2×C4).479(C2×D4), C2.21(C2×C8⋊C22), (C2×C4).318(C2×Q8), C2.51(C2×C22⋊Q8), (C2×C42.C2)⋊31C2, (C2×D4⋊C4).26C2, (C2×C4).836(C4○D4), (C2×C4⋊C4).599C22, SmallGroup(128,1804)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4.Q8
G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=b2d2, 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: 428 in 224 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, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C42.C2, C42.C2, C22×C8, C23×C4, C22×D4, C2×D4⋊C4, C2×C4⋊C8, C2×C4.Q8, C2×C2.D8, D4.Q8, C2×C4×D4, C2×C42.C2, C2×D4.Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C4○D8, C8⋊C22, C22×D4, C22×Q8, C2×C4○D4, D4.Q8, C2×C22⋊Q8, C2×C4○D8, C2×C8⋊C22, C2×D4.Q8
►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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 35)(37 39)(41 43)(45 47)(49 51)(53 55)(57 59)(61 63)
(1 29 5 27)(2 30 6 28)(3 31 7 25)(4 32 8 26)(9 23 15 17)(10 24 16 18)(11 21 13 19)(12 22 14 20)(33 59 39 61)(34 60 40 62)(35 57 37 63)(36 58 38 64)(41 51 47 53)(42 52 48 54)(43 49 45 55)(44 50 46 56)
(1 45 7 41)(2 48 8 44)(3 47 5 43)(4 46 6 42)(9 39 13 35)(10 38 14 34)(11 37 15 33)(12 40 16 36)(17 63 21 59)(18 62 22 58)(19 61 23 57)(20 64 24 60)(25 55 29 51)(26 54 30 50)(27 53 31 49)(28 56 32 52)
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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,35)(37,39)(41,43)(45,47)(49,51)(53,55)(57,59)(61,63), (1,29,5,27)(2,30,6,28)(3,31,7,25)(4,32,8,26)(9,23,15,17)(10,24,16,18)(11,21,13,19)(12,22,14,20)(33,59,39,61)(34,60,40,62)(35,57,37,63)(36,58,38,64)(41,51,47,53)(42,52,48,54)(43,49,45,55)(44,50,46,56), (1,45,7,41)(2,48,8,44)(3,47,5,43)(4,46,6,42)(9,39,13,35)(10,38,14,34)(11,37,15,33)(12,40,16,36)(17,63,21,59)(18,62,22,58)(19,61,23,57)(20,64,24,60)(25,55,29,51)(26,54,30,50)(27,53,31,49)(28,56,32,52)>;
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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,35)(37,39)(41,43)(45,47)(49,51)(53,55)(57,59)(61,63), (1,29,5,27)(2,30,6,28)(3,31,7,25)(4,32,8,26)(9,23,15,17)(10,24,16,18)(11,21,13,19)(12,22,14,20)(33,59,39,61)(34,60,40,62)(35,57,37,63)(36,58,38,64)(41,51,47,53)(42,52,48,54)(43,49,45,55)(44,50,46,56), (1,45,7,41)(2,48,8,44)(3,47,5,43)(4,46,6,42)(9,39,13,35)(10,38,14,34)(11,37,15,33)(12,40,16,36)(17,63,21,59)(18,62,22,58)(19,61,23,57)(20,64,24,60)(25,55,29,51)(26,54,30,50)(27,53,31,49)(28,56,32,52) );
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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,35),(37,39),(41,43),(45,47),(49,51),(53,55),(57,59),(61,63)], [(1,29,5,27),(2,30,6,28),(3,31,7,25),(4,32,8,26),(9,23,15,17),(10,24,16,18),(11,21,13,19),(12,22,14,20),(33,59,39,61),(34,60,40,62),(35,57,37,63),(36,58,38,64),(41,51,47,53),(42,52,48,54),(43,49,45,55),(44,50,46,56)], [(1,45,7,41),(2,48,8,44),(3,47,5,43),(4,46,6,42),(9,39,13,35),(10,38,14,34),(11,37,15,33),(12,40,16,36),(17,63,21,59),(18,62,22,58),(19,61,23,57),(20,64,24,60),(25,55,29,51),(26,54,30,50),(27,53,31,49),(28,56,32,52)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C2×D4.Q8 | C2×D4⋊C4 | C2×C4⋊C8 | C2×C4.Q8 | C2×C2.D8 | D4.Q8 | C2×C4×D4 | C2×C42.C2 | C42 | C22×C4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C2×D4.Q8 ►in GL5(𝔽17)
| 16 | 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 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 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 | 16 | 0 |
| 0 | 0 | 0 | 6 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 10 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 14 | 3 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 11 | 15 |
| 0 | 0 | 0 | 10 | 6 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,6,0,0,0,0,1],[1,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,4,10,0,0,0,0,13],[1,0,0,0,0,0,14,3,0,0,0,3,3,0,0,0,0,0,11,10,0,0,0,15,6] >;
C2×D4.Q8 in GAP, Magma, Sage, TeX
C_2\times D_4.Q_8
% in TeX
G:=Group("C2xD4.Q8"); // GroupNames label
G:=SmallGroup(128,1804);
// by ID
G=gap.SmallGroup(128,1804);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=1,e^2=b^2*d^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