C4^2.59D4 - GroupNames

G = C₄².₅₉D₄ order 128 = 2⁷

41^st non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄².₅₉D₄, (C₄×D₄)⋊₆C₄, C₄⋊Q₈⋊₁₀C₄, C₄⋊₁D₄⋊₈C₄, C₄²⋊₅C₄⋊₁C₂, C₄².₇₆(C₂×C₄), C₂³.₅₀₄(C₂×D₄), (C₂²×C₄).₂₁₆D₄, C₄².₆C₄⋊₃₂C₂, C₂².SD₁₆⋊₂₂C₂, C₄⋊D₄.₁₃₉C₂², C₂²⋊C₈.₁₃₅C₂², C₂².₃₇(C₈⋊C₂²), (C₂²×C₄).₆₃₆C₂³, (C₂×C₄²).₁₇₉C₂², C₂².₂₆C₂⁴.₇C₂, C₂.C₄².₄C₂², C₂.₁₉(C₄²⋊C₂²), C₂.₁₀(C₂³.₃₇D₄), C₂.₁₉(C₂³.C₂³), C₄⋊C₄.₁₄(C₂×C₄), (C₂×D₄).₁₂(C₂×C₄), (C₂×C₄).₁₁₆₀(C₂×D₄), (C₂×C₄).₉₃(C₂²⋊C₄), (C₂×C₄).₁₂₆(C₂²×C₄), C₂².₁₉₀(C₂×C₂²⋊C₄), SmallGroup(128,246)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂×C₄ — C₄².₅₉D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂².₂₆C₂⁴ — C₄².₅₉D₄

Lower central

C₁ — C₂² — C₂×C₄ — C₄².₅₉D₄

Upper central

C₁ — C₂² — C₂×C₄² — C₄².₅₉D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₄².₅₉D₄

Generators and relations for C₄².₅₉D₄
G = < a,b,c,d | a⁴=b⁴=c⁴=1, d²=b^-1, ab=ba, cac^-1=dad^-1=ab², cbc^-1=a²b^-1, bd=db, dcd^-1=b^-1c^-1 >

Subgroups: 308 in 126 conjugacy classes, 44 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂.C₄², C₂.C₄², C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₄×D₄, C₄×D₄, C₄⋊D₄, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂×C₄○D₄, C₂².SD₁₆, C₄²⋊₅C₄, C₄².₆C₄, C₂².₂₆C₂⁴, C₄².₅₉D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₂²⋊C₄, C₈⋊C₂², C₂³.C₂³, C₂³.₃₇D₄, C₄²⋊C₂², C₄².₅₉D₄

Character table of C₄².₅₉D₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	8A	8B	8C	8D
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	-i	-i	1	-1	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	i	i	1	-1	-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	-i	-i	-1	1	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	i	i	-1	1	-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	i	-i	-1	-1	-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	-i	i	-1	-1	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	i	-i	1	1	-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	-i	i	1	1	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 C₈⋊C₂²
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 C₈⋊C₂²
ρ₂₃	4	-4	4	-4	0	0	0	0	-4i	0	0	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	0	-4i	4i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.C₂³
ρ₂₅	4	-4	4	-4	0	0	0	0	4i	0	0	-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	0	4i	-4i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.C₂³

Smallest permutation representation of C₄².₅₉D₄
►On 32 points

Generators in S₃₂

(1 25 15 22)(2 30 16 19)(3 27 9 24)(4 32 10 21)(5 29 11 18)(6 26 12 23)(7 31 13 20)(8 28 14 17)

(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)

(2 14 16 8)(3 13)(4 6 10 12)(7 9)(17 23 28 26)(18 22)(19 32 30 21)(20 31)(24 27)(25 29)

(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)

G:=sub<Sym(32)| (1,25,15,22)(2,30,16,19)(3,27,9,24)(4,32,10,21)(5,29,11,18)(6,26,12,23)(7,31,13,20)(8,28,14,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,14,16,8)(3,13)(4,6,10,12)(7,9)(17,23,28,26)(18,22)(19,32,30,21)(20,31)(24,27)(25,29), (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)>;

G:=Group( (1,25,15,22)(2,30,16,19)(3,27,9,24)(4,32,10,21)(5,29,11,18)(6,26,12,23)(7,31,13,20)(8,28,14,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,14,16,8)(3,13)(4,6,10,12)(7,9)(17,23,28,26)(18,22)(19,32,30,21)(20,31)(24,27)(25,29), (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) );

G=PermutationGroup([[(1,25,15,22),(2,30,16,19),(3,27,9,24),(4,32,10,21),(5,29,11,18),(6,26,12,23),(7,31,13,20),(8,28,14,17)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(2,14,16,8),(3,13),(4,6,10,12),(7,9),(17,23,28,26),(18,22),(19,32,30,21),(20,31),(24,27),(25,29)], [(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)]])

Matrix representation of C₄².₅₉D₄ ►in GL₈(𝔽₁₇)

4	0	0	0	0	0	0	0
0	4	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	1	16	16
0	0	0	0	16	0	1	0
0	0	0	0	0	0	16	16
0	0	0	0	0	0	2	1

1	15	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	4	0
0	0	0	0	0	0	0	4

1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
0	0	0	0	1	0	0	7
0	0	0	0	0	16	0	11
0	0	0	0	0	0	13	13
0	0	0	0	0	0	0	4

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	16	13	7
0	0	0	0	0	1	0	6
0	0	0	0	1	16	0	1
0	0	0	0	0	2	0	16

G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,16,2,0,0,0,0,16,0,16,1],[1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,7,11,13,4],[0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,1,16,2,0,0,0,0,13,0,0,0,0,0,0,0,7,6,1,16] >;

C₄².₅₉D₄ in GAP, Magma, Sage, TeX

C_4^2._{59}D_4

% in TeX

G:=Group("C4^2.59D4");

// GroupNames label

G:=SmallGroup(128,246);

// by ID

G=gap.SmallGroup(128,246);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,520,1123,1018,248,1971]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;

// generators/relations

Export

Character table of C₄².₅₉D₄ in TeX

4	0	0	0	0	0	0	0
0	4	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	1	16	16
0	0	0	0	16	0	1	0
0	0	0	0	0	0	16	16
0	0	0	0	0	0	2	1

1	15	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	4	0
0	0	0	0	0	0	0	4

1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
0	0	0	0	1	0	0	7
0	0	0	0	0	16	0	11
0	0	0	0	0	0	13	13
0	0	0	0	0	0	0	4

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	16	13	7
0	0	0	0	0	1	0	6
0	0	0	0	1	16	0	1
0	0	0	0	0	2	0	16

4	0	0	0	0	0	0	0
0	4	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	1	16	16
0	0	0	0	16	0	1	0
0	0	0	0	0	0	16	16
0	0	0	0	0	0	2	1

1	15	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	4	0
0	0	0	0	0	0	0	4

1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
0	0	0	0	1	0	0	7
0	0	0	0	0	16	0	11
0	0	0	0	0	0	13	13
0	0	0	0	0	0	0	4

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	16	13	7
0	0	0	0	0	1	0	6
0	0	0	0	1	16	0	1
0	0	0	0	0	2	0	16

G = C42.59D4 order 128 = 27

41st non-split extension by C42 of D4 acting via D4/C2=C22

G = C₄².₅₉D₄ order 128 = 2⁷

41^st non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²

4	0	0	0	0	0	0	0
0	4	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	1	16	16
0	0	0	0	16	0	1	0
0	0	0	0	0	0	16	16
0	0	0	0	0	0	2	1

1	15	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	4	0
0	0	0	0	0	0	0	4

1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
0	0	0	0	1	0	0	7
0	0	0	0	0	16	0	11
0	0	0	0	0	0	13	13
0	0	0	0	0	0	0	4

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	16	13	7
0	0	0	0	0	1	0	6
0	0	0	0	1	16	0	1
0	0	0	0	0	2	0	16