direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2xC42:2C2, C42:14C22, C22.22C24, C23.72C23, C24.14C22, (C2xC42):4C2, C4:C4:13C22, (C2xC4).53C23, C22.34(C4oD4), C22:C4.12C22, (C22xC4).60C22, (C2xC4:C4):17C2, C2.11(C2xC4oD4), (C2xC22:C4).12C2, SmallGroup(64,209)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2xC42:2C2
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >
Subgroups: 177 in 123 conjugacy classes, 81 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2xC4, C2xC4, C23, C23, C23, C42, C22:C4, C4:C4, C22xC4, C24, C2xC42, C2xC22:C4, C2xC4:C4, C42:2C2, C2xC42:2C2
Quotients: C1, C2, C22, C23, C4oD4, C24, C42:2C2, C2xC4oD4, C2xC42:2C2
Character table of C2xC42:2C2
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ11 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ13 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ15 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ19 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ22 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ24 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ25 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ26 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ27 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 16)(6 13)(7 14)(8 15)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 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)
(1 17 13 31)(2 18 14 32)(3 19 15 29)(4 20 16 30)(5 24 28 10)(6 21 25 11)(7 22 26 12)(8 23 27 9)
(1 27)(2 5)(3 25)(4 7)(6 15)(8 13)(9 29)(10 20)(11 31)(12 18)(14 28)(16 26)(17 21)(19 23)(22 32)(24 30)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,16)(6,13)(7,14)(8,15)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (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), (1,17,13,31)(2,18,14,32)(3,19,15,29)(4,20,16,30)(5,24,28,10)(6,21,25,11)(7,22,26,12)(8,23,27,9), (1,27)(2,5)(3,25)(4,7)(6,15)(8,13)(9,29)(10,20)(11,31)(12,18)(14,28)(16,26)(17,21)(19,23)(22,32)(24,30)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,16)(6,13)(7,14)(8,15)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (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), (1,17,13,31)(2,18,14,32)(3,19,15,29)(4,20,16,30)(5,24,28,10)(6,21,25,11)(7,22,26,12)(8,23,27,9), (1,27)(2,5)(3,25)(4,7)(6,15)(8,13)(9,29)(10,20)(11,31)(12,18)(14,28)(16,26)(17,21)(19,23)(22,32)(24,30) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,16),(6,13),(7,14),(8,15),(9,19),(10,20),(11,17),(12,18),(21,31),(22,32),(23,29),(24,30)], [(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)], [(1,17,13,31),(2,18,14,32),(3,19,15,29),(4,20,16,30),(5,24,28,10),(6,21,25,11),(7,22,26,12),(8,23,27,9)], [(1,27),(2,5),(3,25),(4,7),(6,15),(8,13),(9,29),(10,20),(11,31),(12,18),(14,28),(16,26),(17,21),(19,23),(22,32),(24,30)]])
C2xC42:2C2 is a maximal subgroup of
C42.372D4 C24.203C23 C23.255C24 C24.230C23 C24.286C23 C23.367C24 C23.368C24 C23.369C24 C24.295C23 C23.379C24 C23.380C24 C24.573C23 C23.396C24 C23.412C24 C23.419C24 C24.311C23 C24.326C23 C24.327C23 C24.331C23 C24.332C23 C42:23D4 C42:24D4 C42.184D4 C42:30D4 C42.192D4 C42:32D4 C42.198D4 C23.585C24 C23.589C24 C23.591C24 C23.595C24 C24.405C23 C23.602C24 C23.605C24 C24.413C23 C23.615C24 C23.616C24 C23.618C24 C23.621C24 C23.622C24 C24.418C23 C23.625C24 C23.627C24 C42:33D4 C42.200D4 C42:35D4 C42:43D4 C43:13C2 C22.110C25 C22.149C25 C22.153C25
C2xC42:2C2 is a maximal quotient of
C23.301C24 C23.380C24 C24.573C23 C24.577C23 C24.304C23 C23.395C24 C23.396C24 C23.397C24 C23.410C24 C23.411C24 C23.412C24 C23.413C24 C23.414C24 C23.543C24 C23.544C24 C23.545C24 C42:43D4 C43:13C2 C42:15Q8
Matrix representation of C2xC42:2C2 ►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 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 3 | 4 |
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],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,4,1],[1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,3,2],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,3,0,0,0,0,4] >;
C2xC42:2C2 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_2C_2
% in TeX
G:=Group("C2xC4^2:2C2"); // GroupNames label
G:=SmallGroup(64,209);
// by ID
G=gap.SmallGroup(64,209);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,86]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,d*c*d=b^2*c^-1>;
// generators/relations
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