direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4xC4wrC2, C43:6C2, D4:1C42, Q8:1C42, C42.458D4, (C4xD4):9C4, (C4xQ8):9C4, C42:36(C2xC4), C4.161(C4xD4), C4.1(C2xC42), C22.26(C4xD4), C4o2(C42:6C4), C42:6C4:34C2, (C4xM4(2)):21C2, M4(2):12(C2xC4), (C22xC4).670D4, C23.541(C2xD4), C4.1(C42:C2), C42o(C42:6C4), (C2xC42).1045C22, (C22xC4).1305C23, C42:C2.261C22, (C2xM4(2)).306C22, C2.5(C2xC4wrC2), C42o(C2xC4wrC2), (C4xC4oD4).5C2, (C2xC4wrC2).16C2, C4:C4.187(C2xC4), C4oD4.13(C2xC4), C2.16(C4xC22:C4), (C2xD4).196(C2xC4), (C2xC4).1499(C2xD4), (C2xQ8).179(C2xC4), (C2xC4).536(C4oD4), (C2xC4).348(C22xC4), (C2xC4).399(C22:C4), (C2xC4oD4).250C22, C22.117(C2xC22:C4), 2-Sylow(GL(3,5)), SmallGroup(128,490)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4xC4wrC2
G = < a,b,c,d | a4=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Subgroups: 300 in 180 conjugacy classes, 80 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C4xC8, C8:C4, C4wrC2, C2xC42, C2xC42, C2xC42, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C2xM4(2), C2xC4oD4, C42:6C4, C43, C4xM4(2), C2xC4wrC2, C4xC4oD4, C4xC4wrC2
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C42, C22:C4, C22xC4, C2xD4, C4oD4, C4wrC2, C2xC42, C2xC22:C4, C42:C2, C4xD4, C4xC22:C4, C2xC4wrC2, C4xC4wrC2
►On 32 points
(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 8 23 9)(2 5 24 10)(3 6 21 11)(4 7 22 12)(13 32 27 18)(14 29 28 19)(15 30 25 20)(16 31 26 17)
(1 18)(2 19)(3 20)(4 17)(5 28)(6 25)(7 26)(8 27)(9 13)(10 14)(11 15)(12 16)(21 30)(22 31)(23 32)(24 29)
(1 8 23 9)(2 5 24 10)(3 6 21 11)(4 7 22 12)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)
G:=sub<Sym(32)| (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), (1,8,23,9)(2,5,24,10)(3,6,21,11)(4,7,22,12)(13,32,27,18)(14,29,28,19)(15,30,25,20)(16,31,26,17), (1,18)(2,19)(3,20)(4,17)(5,28)(6,25)(7,26)(8,27)(9,13)(10,14)(11,15)(12,16)(21,30)(22,31)(23,32)(24,29), (1,8,23,9)(2,5,24,10)(3,6,21,11)(4,7,22,12)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)>;
G:=Group( (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), (1,8,23,9)(2,5,24,10)(3,6,21,11)(4,7,22,12)(13,32,27,18)(14,29,28,19)(15,30,25,20)(16,31,26,17), (1,18)(2,19)(3,20)(4,17)(5,28)(6,25)(7,26)(8,27)(9,13)(10,14)(11,15)(12,16)(21,30)(22,31)(23,32)(24,29), (1,8,23,9)(2,5,24,10)(3,6,21,11)(4,7,22,12)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30) );
G=PermutationGroup([[(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)], [(1,8,23,9),(2,5,24,10),(3,6,21,11),(4,7,22,12),(13,32,27,18),(14,29,28,19),(15,30,25,20),(16,31,26,17)], [(1,18),(2,19),(3,20),(4,17),(5,28),(6,25),(7,26),(8,27),(9,13),(10,14),(11,15),(12,16),(21,30),(22,31),(23,32),(24,29)], [(1,8,23,9),(2,5,24,10),(3,6,21,11),(4,7,22,12),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4AH | 4AI | ··· | 4AN | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4oD4 | C4wrC2 |
| kernel | C4xC4wrC2 | C42:6C4 | C43 | C4xM4(2) | C2xC4wrC2 | C4xC4oD4 | C4wrC2 | C4xD4 | C4xQ8 | C42 | C22xC4 | C2xC4 | C4 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 4 | 4 | 2 | 2 | 4 | 16 |
Matrix representation of C4xC4wrC2 ►in GL3(F17) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 13 |
| 16 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 16 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [4,0,0,0,4,0,0,0,4],[1,0,0,0,4,0,0,0,13],[16,0,0,0,0,1,0,1,0],[16,0,0,0,4,0,0,0,16] >;
C4xC4wrC2 in GAP, Magma, Sage, TeX
C_4\times C_4\wr C_2
% in TeX
G:=Group("C4xC4wrC2"); // GroupNames label
G:=SmallGroup(128,490);
// by ID
G=gap.SmallGroup(128,490);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,4037]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations