direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4xC22xC4, C42:20C23, C22.8C25, C25.94C22, C23.103C24, C24.602C23, (C24xC4):5C2, C4:1(C23xC4), C4:C4:23C23, C24:16(C2xC4), C2.4(C24xC4), C2.3(D4xC23), C22:1(C23xC4), C23:6(C22xC4), C22:C4:21C23, (C2xC4).155C24, (C2xC42):87C22, (C22xC42):19C2, (C23xC4):57C22, (C22xC4):22C23, (D4xC23).22C2, C23.888(C2xD4), (C2xD4).494C23, C23.375(C4oD4), C22.156(C22xD4), (C22xD4).611C22, C4o(C2xC4xD4), (C2xC4)o2(C4xD4), C4:C4o(C23xC4), C4:C4o3(C22xC4), C4o2(C22xC4:C4), (C22xC4)o(C4xD4), (C2xC4)o(D4xC23), (C2xD4)o(C23xC4), (C2xC4):9(C22xC4), C22:C4o(C23xC4), (C22xC4:C4):49C2, (C22xC4):47(C2xC4), C22:C4o3(C22xC4), C4o2(C22xC22:C4), C2.2(C22xC4oD4), (C2xC4:C4):144C22, (C23xC4)o(C22xD4), (C22xC4)o(D4xC23), (C23xC4)o(D4xC23), (C22xC4)o2(C22xD4), (C22xC22:C4):35C2, (C2xC22:C4):90C22, C22.143(C2xC4oD4), (C2xC4)o2(C2xC4xD4), (C2xC4)o5(C2xC4:C4), (C22xC4)o(C2xC4xD4), (C2xC4:C4)o(C23xC4), (C2xC4)o4(C2xC22:C4), (C2xC4)o2(C22xC4:C4), (C22xC4)o3(C2xC4:C4), (C2xC22:C4)o(C23xC4), (C23xC4)o(C22xC4:C4), (C22xC4)o3(C2xC22:C4), (C2xC4)o2(C22xC22:C4), (C22xC4)o2(C22xC4:C4), (C23xC4)o(C22xC22:C4), (C22xC4)o2(C22xC22:C4), SmallGroup(128,2154)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4xC22xC4
G = < a,b,c,d,e | a2=b2=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1756 in 1264 conjugacy classes, 772 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, C23, C23, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C24, C24, C24, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C23xC4, C23xC4, C23xC4, C22xD4, C25, C22xC42, C22xC22:C4, C22xC4:C4, C2xC4xD4, C24xC4, D4xC23, D4xC22xC4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C23xC4, C22xD4, C2xC4oD4, C25, C2xC4xD4, C24xC4, D4xC23, C22xC4oD4, D4xC22xC4
►On 64 points
(1 21)(2 22)(3 23)(4 24)(5 11)(6 12)(7 9)(8 10)(13 44)(14 41)(15 42)(16 43)(17 29)(18 30)(19 31)(20 32)(25 36)(26 33)(27 34)(28 35)(37 50)(38 51)(39 52)(40 49)(45 57)(46 58)(47 59)(48 60)(53 61)(54 62)(55 63)(56 64)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 63)(26 64)(27 61)(28 62)(29 41)(30 42)(31 43)(32 44)(33 56)(34 53)(35 54)(36 55)(37 57)(38 58)(39 59)(40 60)(45 50)(46 51)(47 52)(48 49)
(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 45 31 63)(2 46 32 64)(3 47 29 61)(4 48 30 62)(5 37 16 36)(6 38 13 33)(7 39 14 34)(8 40 15 35)(9 52 41 27)(10 49 42 28)(11 50 43 25)(12 51 44 26)(17 53 23 59)(18 54 24 60)(19 55 21 57)(20 56 22 58)
(1 19)(2 20)(3 17)(4 18)(5 43)(6 44)(7 41)(8 42)(9 14)(10 15)(11 16)(12 13)(21 31)(22 32)(23 29)(24 30)(25 36)(26 33)(27 34)(28 35)(37 50)(38 51)(39 52)(40 49)(45 57)(46 58)(47 59)(48 60)(53 61)(54 62)(55 63)(56 64)
G:=sub<Sym(64)| (1,21)(2,22)(3,23)(4,24)(5,11)(6,12)(7,9)(8,10)(13,44)(14,41)(15,42)(16,43)(17,29)(18,30)(19,31)(20,32)(25,36)(26,33)(27,34)(28,35)(37,50)(38,51)(39,52)(40,49)(45,57)(46,58)(47,59)(48,60)(53,61)(54,62)(55,63)(56,64), (1,11)(2,12)(3,9)(4,10)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,63)(26,64)(27,61)(28,62)(29,41)(30,42)(31,43)(32,44)(33,56)(34,53)(35,54)(36,55)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49), (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), (1,45,31,63)(2,46,32,64)(3,47,29,61)(4,48,30,62)(5,37,16,36)(6,38,13,33)(7,39,14,34)(8,40,15,35)(9,52,41,27)(10,49,42,28)(11,50,43,25)(12,51,44,26)(17,53,23,59)(18,54,24,60)(19,55,21,57)(20,56,22,58), (1,19)(2,20)(3,17)(4,18)(5,43)(6,44)(7,41)(8,42)(9,14)(10,15)(11,16)(12,13)(21,31)(22,32)(23,29)(24,30)(25,36)(26,33)(27,34)(28,35)(37,50)(38,51)(39,52)(40,49)(45,57)(46,58)(47,59)(48,60)(53,61)(54,62)(55,63)(56,64)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,11)(6,12)(7,9)(8,10)(13,44)(14,41)(15,42)(16,43)(17,29)(18,30)(19,31)(20,32)(25,36)(26,33)(27,34)(28,35)(37,50)(38,51)(39,52)(40,49)(45,57)(46,58)(47,59)(48,60)(53,61)(54,62)(55,63)(56,64), (1,11)(2,12)(3,9)(4,10)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,63)(26,64)(27,61)(28,62)(29,41)(30,42)(31,43)(32,44)(33,56)(34,53)(35,54)(36,55)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49), (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), (1,45,31,63)(2,46,32,64)(3,47,29,61)(4,48,30,62)(5,37,16,36)(6,38,13,33)(7,39,14,34)(8,40,15,35)(9,52,41,27)(10,49,42,28)(11,50,43,25)(12,51,44,26)(17,53,23,59)(18,54,24,60)(19,55,21,57)(20,56,22,58), (1,19)(2,20)(3,17)(4,18)(5,43)(6,44)(7,41)(8,42)(9,14)(10,15)(11,16)(12,13)(21,31)(22,32)(23,29)(24,30)(25,36)(26,33)(27,34)(28,35)(37,50)(38,51)(39,52)(40,49)(45,57)(46,58)(47,59)(48,60)(53,61)(54,62)(55,63)(56,64) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,11),(6,12),(7,9),(8,10),(13,44),(14,41),(15,42),(16,43),(17,29),(18,30),(19,31),(20,32),(25,36),(26,33),(27,34),(28,35),(37,50),(38,51),(39,52),(40,49),(45,57),(46,58),(47,59),(48,60),(53,61),(54,62),(55,63),(56,64)], [(1,11),(2,12),(3,9),(4,10),(5,21),(6,22),(7,23),(8,24),(13,20),(14,17),(15,18),(16,19),(25,63),(26,64),(27,61),(28,62),(29,41),(30,42),(31,43),(32,44),(33,56),(34,53),(35,54),(36,55),(37,57),(38,58),(39,59),(40,60),(45,50),(46,51),(47,52),(48,49)], [(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)], [(1,45,31,63),(2,46,32,64),(3,47,29,61),(4,48,30,62),(5,37,16,36),(6,38,13,33),(7,39,14,34),(8,40,15,35),(9,52,41,27),(10,49,42,28),(11,50,43,25),(12,51,44,26),(17,53,23,59),(18,54,24,60),(19,55,21,57),(20,56,22,58)], [(1,19),(2,20),(3,17),(4,18),(5,43),(6,44),(7,41),(8,42),(9,14),(10,15),(11,16),(12,13),(21,31),(22,32),(23,29),(24,30),(25,36),(26,33),(27,34),(28,35),(37,50),(38,51),(39,52),(40,49),(45,57),(46,58),(47,59),(48,60),(53,61),(54,62),(55,63),(56,64)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2AE | 4A | ··· | 4P | 4Q | ··· | 4AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4oD4 |
| kernel | D4xC22xC4 | C22xC42 | C22xC22:C4 | C22xC4:C4 | C2xC4xD4 | C24xC4 | D4xC23 | C22xD4 | C22xC4 | C23 |
| # reps | 1 | 1 | 2 | 1 | 24 | 2 | 1 | 32 | 8 | 8 |
Matrix representation of D4xC22xC4 ►in GL5(F5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,4,0,0,0,2,4],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4,1,0,0,0,0,1] >;
D4xC22xC4 in GAP, Magma, Sage, TeX
D_4\times C_2^2\times C_4
% in TeX
G:=Group("D4xC2^2xC4"); // GroupNames label
G:=SmallGroup(128,2154);
// by ID
G=gap.SmallGroup(128,2154);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,352]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations