direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2xC4.10D4, M4(2).7C22, C4.45(C2xD4), (C2xQ8).6C4, (C2xC4).121D4, (C22xC4).5C4, (C2xC4).2C23, C23.30(C2xC4), (C22xQ8).3C2, C4.11(C22:C4), C22.9(C22xC4), (C2xQ8).38C22, (C2xM4(2)).11C2, (C22xC4).32C22, C22.31(C22:C4), (C2xC4).6(C2xC4), C2.15(C2xC22:C4), SmallGroup(64,93)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2xC4.10D4
G = < a,b,c,d | a2=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Subgroups: 105 in 73 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, C2xC4, Q8, C23, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xQ8, C2xQ8, C4.10D4, C2xM4(2), C22xQ8, C2xC4.10D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4.10D4, C2xC22:C4, C2xC4.10D4
Character table of C2xC4.10D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | trivial |
ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | i | -i | i | -i | linear of order 4 |
ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | -i | -i | i | i | -i | i | i | linear of order 4 |
ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | i | -i | -i | -i | i | i | -i | i | linear of order 4 |
ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -i | i | -i | i | linear of order 4 |
ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | i | i | i | -i | -i | i | -i | -i | linear of order 4 |
ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | i | i | i | -i | -i | i | -i | linear of order 4 |
ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 20 13 24)(10 17 14 21)(11 22 15 18)(12 19 16 23)
(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 20 31 9 5 24 27 13)(2 16 28 19 6 12 32 23)(3 18 25 15 7 22 29 11)(4 14 30 17 8 10 26 21)
G:=sub<Sym(32)| (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,20,13,24)(10,17,14,21)(11,22,15,18)(12,19,16,23), (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), (1,20,31,9,5,24,27,13)(2,16,28,19,6,12,32,23)(3,18,25,15,7,22,29,11)(4,14,30,17,8,10,26,21)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,20,13,24)(10,17,14,21)(11,22,15,18)(12,19,16,23), (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), (1,20,31,9,5,24,27,13)(2,16,28,19,6,12,32,23)(3,18,25,15,7,22,29,11)(4,14,30,17,8,10,26,21) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(1,27,5,31),(2,32,6,28),(3,29,7,25),(4,26,8,30),(9,20,13,24),(10,17,14,21),(11,22,15,18),(12,19,16,23)], [(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)], [(1,20,31,9,5,24,27,13),(2,16,28,19,6,12,32,23),(3,18,25,15,7,22,29,11),(4,14,30,17,8,10,26,21)]])
C2xC4.10D4 is a maximal subgroup of
(C2xQ8).Q8 C4.C22wrC2 (C23xC4).C4 2- 1+4:2C4 C42.97D4 (C2xQ8).211D4 C4.10D4:2C4 C4:Q8:15C4 C4.4D4:13C4 M4(2).45D4 M4(2).46D4 M4(2).49D4 C42.7D4 M4(2).50D4 C4.10D4:3C4 C42.114D4 C42.115D4 M4(2).31D4 M4(2).33D4 C42.129D4 C42.130D4 M4(2).5D4 M4(2).6D4 M4(2).7D4 C4:C4.97D4 C4:C4.98D4 M4(2).9D4 M4(2).11D4 M4(2).13D4 M4(2).15D4 (C2xC8).6D4 C4:Q8.C4 M4(2).25C23 M4(2).C23 M4(2).38D4 (C2xQ8).7F5
C2xC4.10D4 is a maximal quotient of
C42.394D4 (C2xC4):M4(2) C42.44D4 C42.396D4 C24.45(C2xC4) C42.406D4 C42.408D4 C42.68D4 C42.71D4 C42.74D4 C42.412D4 C42.414D4 C42.416D4 C42.81D4 C42.83D4 C42.88D4 C4.C22wrC2 C42.97D4 (C22xC4).275D4 M4(2).45D4 C42.114D4 M4(2):8Q8 (C2xQ8).7F5
Matrix representation of C2xC4.10D4 ►in GL6(F17)
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 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 | 0 | 0 | 1 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 2 | 0 | 0 |
0 | 0 | 16 | 1 | 0 | 0 |
0 | 0 | 16 | 1 | 0 | 1 |
0 | 0 | 0 | 1 | 16 | 0 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 2 | 0 |
0 | 0 | 0 | 0 | 1 | 16 |
0 | 0 | 0 | 16 | 1 | 0 |
0 | 0 | 16 | 0 | 1 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 8 | 0 | 3 | 15 |
0 | 0 | 7 | 0 | 11 | 9 |
0 | 0 | 7 | 10 | 10 | 16 |
0 | 0 | 8 | 16 | 10 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,16,0,0,0,2,1,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,16,0,0,0,2,1,1,1,0,0,0,16,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,8,7,7,8,0,0,0,0,10,16,0,0,3,11,10,10,0,0,15,9,16,16] >;
C2xC4.10D4 in GAP, Magma, Sage, TeX
C_2\times C_4._{10}D_4
% in TeX
G:=Group("C2xC4.10D4");
// GroupNames label
G:=SmallGroup(64,93);
// by ID
G=gap.SmallGroup(64,93);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,963,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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