p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.70D4, C42.151C23, C42.92(C2×C4), C4.4D4.9C4, (C22×D4).9C4, C8⋊C4⋊27C22, (C22×Q8).8C4, (C22×C4).228D4, C42.6C4⋊37C2, (C2×C42).195C22, C23.179(C22⋊C4), C42.C22⋊11C2, C4.4D4.117C22, C22.17(C4.D4), C2.33(C42⋊C22), (C2×D4).24(C2×C4), (C2×Q8).24(C2×C4), (C2×C4).1179(C2×D4), (C2×C4.4D4).4C2, C2.13(C2×C4.D4), (C2×C4).96(C22⋊C4), (C2×C4).145(C22×C4), (C22×C4).217(C2×C4), C22.209(C2×C22⋊C4), SmallGroup(128,265)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.70D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 308 in 124 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C22×D4, C22×Q8, C42.C22, C42.6C4, C2×C4.4D4, C42.70D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C2×C4.D4, C42⋊C22, C42.70D4
Character table of C42.70D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
ρ9 | 1 | 1 | 1 | 1 | -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 | 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 | 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 | 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 | 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 | 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 | 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 | 1 | 1 | -1 | 1 | i | -i | -i | -i | i | i | -i | i | linear of order 4 |
ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 4 | 4 | -4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
ρ22 | 4 | 4 | -4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
(1 30 24 10)(2 15 17 27)(3 32 18 12)(4 9 19 29)(5 26 20 14)(6 11 21 31)(7 28 22 16)(8 13 23 25)
(1 12 20 28)(2 9 21 25)(3 14 22 30)(4 11 23 27)(5 16 24 32)(6 13 17 29)(7 10 18 26)(8 15 19 31)
(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 15 12 19 20 31 28 8)(2 22 9 30 21 3 25 14)(4 5 11 16 23 24 27 32)(6 18 13 26 17 7 29 10)
G:=sub<Sym(32)| (1,30,24,10)(2,15,17,27)(3,32,18,12)(4,9,19,29)(5,26,20,14)(6,11,21,31)(7,28,22,16)(8,13,23,25), (1,12,20,28)(2,9,21,25)(3,14,22,30)(4,11,23,27)(5,16,24,32)(6,13,17,29)(7,10,18,26)(8,15,19,31), (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), (1,15,12,19,20,31,28,8)(2,22,9,30,21,3,25,14)(4,5,11,16,23,24,27,32)(6,18,13,26,17,7,29,10)>;
G:=Group( (1,30,24,10)(2,15,17,27)(3,32,18,12)(4,9,19,29)(5,26,20,14)(6,11,21,31)(7,28,22,16)(8,13,23,25), (1,12,20,28)(2,9,21,25)(3,14,22,30)(4,11,23,27)(5,16,24,32)(6,13,17,29)(7,10,18,26)(8,15,19,31), (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), (1,15,12,19,20,31,28,8)(2,22,9,30,21,3,25,14)(4,5,11,16,23,24,27,32)(6,18,13,26,17,7,29,10) );
G=PermutationGroup([[(1,30,24,10),(2,15,17,27),(3,32,18,12),(4,9,19,29),(5,26,20,14),(6,11,21,31),(7,28,22,16),(8,13,23,25)], [(1,12,20,28),(2,9,21,25),(3,14,22,30),(4,11,23,27),(5,16,24,32),(6,13,17,29),(7,10,18,26),(8,15,19,31)], [(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)], [(1,15,12,19,20,31,28,8),(2,22,9,30,21,3,25,14),(4,5,11,16,23,24,27,32),(6,18,13,26,17,7,29,10)]])
Matrix representation of C42.70D4 ►in GL8(𝔽17)
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
3 | 10 | 14 | 10 | 0 | 0 | 0 | 0 |
10 | 14 | 10 | 3 | 0 | 0 | 0 | 0 |
3 | 7 | 14 | 7 | 0 | 0 | 0 | 0 |
7 | 14 | 7 | 3 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
3 | 10 | 14 | 10 | 0 | 0 | 0 | 0 |
7 | 3 | 7 | 14 | 0 | 0 | 0 | 0 |
3 | 7 | 14 | 7 | 0 | 0 | 0 | 0 |
10 | 3 | 10 | 14 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[3,10,3,7,0,0,0,0,10,14,7,14,0,0,0,0,14,10,14,7,0,0,0,0,10,3,7,3,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[3,7,3,10,0,0,0,0,10,3,7,3,0,0,0,0,14,7,14,10,0,0,0,0,10,14,7,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.70D4 in GAP, Magma, Sage, TeX
C_4^2._{70}D_4
% in TeX
G:=Group("C4^2.70D4");
// GroupNames label
G:=SmallGroup(128,265);
// by ID
G=gap.SmallGroup(128,265);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations
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