direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2xM5(2):C2, C23.36SD16, M5(2):15C22, C4.64(C2xD8), C8.96(C2xD4), (C2xD8).14C4, D8.12(C2xC4), (C2xC4).142D8, (C2xC8).123D4, C8.8(C22xC4), (C2xC4).53SD16, C8.10(C22:C4), (C2xM5(2)):17C2, (C2xC8).226C23, C8.C4:9C22, (C22xD8).14C2, (C22xC4).336D4, C4.26(D4:C4), (C2xD8).153C22, C22.16(C2xSD16), (C22xC8).235C22, C22.33(D4:C4), (C2xC8).85(C2xC4), (C2xC4).271(C2xD4), C4.58(C2xC22:C4), (C2xC8.C4):19C2, C2.36(C2xD4:C4), (C2xC4).274(C22:C4), SmallGroup(128,878)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2xM5(2):C2
G = < a,b,c,d | a2=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b9, dbd=b3c, cd=dc >
Subgroups: 372 in 128 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C8, C2xC4, D4, C23, C23, C16, C2xC8, C2xC8, M4(2), D8, D8, C22xC4, C2xD4, C24, C8.C4, C8.C4, C2xC16, M5(2), M5(2), C22xC8, C2xM4(2), C2xD8, C2xD8, C22xD4, M5(2):C2, C2xC8.C4, C2xM5(2), C22xD8, C2xM5(2):C2
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, D8, SD16, C22xC4, C2xD4, D4:C4, C2xC22:C4, C2xD8, C2xSD16, M5(2):C2, C2xD4:C4, C2xM5(2):C2
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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)(2 26)(3 19)(4 28)(5 21)(6 30)(7 23)(8 32)(9 25)(10 18)(11 27)(12 20)(13 29)(14 22)(15 31)(16 24)
(1 5)(2 32)(4 30)(6 28)(7 15)(8 26)(9 13)(10 24)(12 22)(14 20)(16 18)(17 21)(23 31)(25 29)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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)(2,26)(3,19)(4,28)(5,21)(6,30)(7,23)(8,32)(9,25)(10,18)(11,27)(12,20)(13,29)(14,22)(15,31)(16,24), (1,5)(2,32)(4,30)(6,28)(7,15)(8,26)(9,13)(10,24)(12,22)(14,20)(16,18)(17,21)(23,31)(25,29)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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)(2,26)(3,19)(4,28)(5,21)(6,30)(7,23)(8,32)(9,25)(10,18)(11,27)(12,20)(13,29)(14,22)(15,31)(16,24), (1,5)(2,32)(4,30)(6,28)(7,15)(8,26)(9,13)(10,24)(12,22)(14,20)(16,18)(17,21)(23,31)(25,29) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(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),(2,26),(3,19),(4,28),(5,21),(6,30),(7,23),(8,32),(9,25),(10,18),(11,27),(12,20),(13,29),(14,22),(15,31),(16,24)], [(1,5),(2,32),(4,30),(6,28),(7,15),(8,26),(9,13),(10,24),(12,22),(14,20),(16,18),(17,21),(23,31),(25,29)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | SD16 | M5(2):C2 |
| kernel | C2xM5(2):C2 | M5(2):C2 | C2xC8.C4 | C2xM5(2) | C22xD8 | C2xD8 | C2xC8 | C22xC4 | C2xC4 | C2xC4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C2xM5(2):C2 ►in GL6(F17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 11 | 14 | 0 | 0 | 0 | 0 |
| 3 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 2 | 0 |
| 0 | 0 | 14 | 0 | 16 | 1 |
| 0 | 0 | 13 | 14 | 12 | 0 |
| 0 | 0 | 14 | 3 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 16 | 0 |
| 0 | 0 | 1 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 | 14 | 3 |
| 0 | 0 | 14 | 0 | 3 | 3 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[11,3,0,0,0,0,14,13,0,0,0,0,0,0,5,14,13,14,0,0,0,0,14,3,0,0,2,16,12,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,12,0,0,0,0,0,1,0,0,0,0,0,0,16,1,2,14,0,0,0,1,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;
C2xM5(2):C2 in GAP, Magma, Sage, TeX
C_2\times M_5(2)\rtimes C_2
% in TeX
G:=Group("C2xM5(2):C2"); // GroupNames label
G:=SmallGroup(128,878);
// by ID
G=gap.SmallGroup(128,878);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1466,136,1411,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^9,d*b*d=b^3*c,c*d=d*c>;
// generators/relations