(C2xC4).SD16 - GroupNames

G = (C₂xC₄).SD₁₆ order 128 = 2⁷

7^th non-split extension by C₂xC₄ of SD₁₆ acting via SD₁₆/C₂=D₄

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

Aliases: C₄:C₄.₁₄D₄, (C₂xQ₈).₁₇D₄, (C₂xC₄).₇SD₁₆, (C₂²xC₄).₅₁D₄, C₂.₉(Q₈:D₄), C₂³.₅₂₈(C₂xD₄), (C₂²xC₄).₁₇C₂³, C₂².₁₅(C₂xSD₁₆), C₂²:Q₈.₁₁C₂², C₂².₁₃₈C₂²wrC₂, C₂²:C₈.₁₁₆C₂², C₂³.₃₁D₄.₄C₂, C₂³.₄₇D₄.₄C₂, C₂.₈(C₂³.₇D₄), C₂.₁₁(D₄.₁₀D₄), C₂².₃₂(C₈.C₂²), C₂.C₄².₂₄C₂², C₂³.₈₁C₂³.₂C₂, C₂³.₄₁C₂³.₂C₂, C₂².M₄(2).₉C₂, (C₂xC₄).₂₀₆(C₂xD₄), (C₂xC₄:C₄).₂₂C₂², SmallGroup(128,343)

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

Derived series

C₁ — C₂²xC₄ — (C₂xC₄).SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₄ — C₂xC₄:C₄ — C₂³.₄₁C₂³ — (C₂xC₄).SD₁₆

Lower central

C₁ — C₂² — C₂²xC₄ — (C₂xC₄).SD₁₆

Upper central

C₁ — C₂² — C₂²xC₄ — (C₂xC₄).SD₁₆

Jennings

C₁ — C₂ — C₂² — C₂²xC₄ — (C₂xC₄).SD₁₆

Generators and relations for (C₂xC₄).SD₁₆
G = < a,b,c,d | a²=b⁴=c⁸=1, d²=b², dbd^-1=ab=ba, cac^-1=ab², ad=da, cbc^-1=ab^-1, dcd^-1=b²c³ >

Subgroups: 228 in 105 conjugacy classes, 34 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂xC₄, C₂xC₄, Q₈, C₂³, C₄², C₂²:C₄, C₄:C₄, C₄:C₄, C₂xC₈, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₂.C₄², C₂.C₄², C₂²:C₈, Q₈:C₄, C₄.Q₈, C₂xC₄:C₄, C₂xC₄:C₄, C₄²:C₂, C₂²:Q₈, C₂²:Q₈, C₄².C₂, C₄:Q₈, C₂².M₄(2), C₂³.₃₁D₄, C₂³.₈₁C₂³, C₂³.₄₇D₄, C₂³.₄₁C₂³, (C₂xC₄).SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂xD₄, C₂²wrC₂, C₂xSD₁₆, C₈.C₂², Q₈:D₄, D₄.₁₀D₄, C₂³.₇D₄, (C₂xC₄).SD₁₆

Character table of (C₂xC₄).SD₁₆

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	8A	8B	8C	8D
size	1	1	1	1	2	2	4	4	4	4	8	8	8	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	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	-2	-2	0	0	-2	2	0	0	-2	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	2	2	-2	-2	0	0	0	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	0	0	2	-2	0	2	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	0	0	2	-2	0	-2	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	-2	0	0	-2	2	0	0	2	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₁₆	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₇	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₁₈	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₉	4	-4	-4	4	0	0	0	0	0	0	2	0	0	0	0	0	0	0	-2	0	0	0	0	symplectic lifted from D₄.₁₀D₄, Schur index 2
ρ₂₀	4	4	-4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₁	4	-4	-4	4	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	2	0	0	0	0	symplectic lifted from D₄.₁₀D₄, Schur index 2
ρ₂₂	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	0	0	0	0	complex lifted from C₂³.₇D₄
ρ₂₃	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	0	0	0	0	complex lifted from C₂³.₇D₄

Smallest permutation representation of (C₂xC₄).SD₁₆
►On 32 points

Generators in S₃₂

(2 28)(4 30)(6 32)(8 26)(10 20)(12 22)(14 24)(16 18)

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

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

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

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

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

Matrix representation of (C₂xC₄).SD₁₆ ►in GL₆(F₁₇)

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	0	0	16	0
0	0	16	16	0	16

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	16	0	0
0	0	16	10	0	0
0	0	1	1	8	2
0	0	10	0	10	9

5	5	0	0	0	0
12	5	0	0	0	0
0	0	16	16	9	15
0	0	10	10	11	3
0	0	1	7	0	0
0	0	0	1	7	8

1	0	0	0	0	0
0	16	0	0	0	0
0	0	10	1	0	0
0	0	1	7	0	0
0	0	7	7	6	14
0	0	0	10	1	11

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,16,1,10,0,0,16,10,1,0,0,0,0,0,8,10,0,0,0,0,2,9],[5,12,0,0,0,0,5,5,0,0,0,0,0,0,16,10,1,0,0,0,16,10,7,1,0,0,9,11,0,7,0,0,15,3,0,8],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,10,1,7,0,0,0,1,7,7,10,0,0,0,0,6,1,0,0,0,0,14,11] >;

(C₂xC₄).SD₁₆ in GAP, Magma, Sage, TeX

(C_2\times C_4).{\rm SD}_{16}

% in TeX

G:=Group("(C2xC4).SD16");

// GroupNames label

G:=SmallGroup(128,343);

// by ID

G=gap.SmallGroup(128,343);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,352,1123,570,521,136,1411]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2,d*b*d^-1=a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=a*b^-1,d*c*d^-1=b^2*c^3>;

// generators/relations

Export

Character table of (C₂xC₄).SD₁₆ in TeX

G = (C2xC4).SD16 order 128 = 27

7th non-split extension by C2xC4 of SD16 acting via SD16/C2=D4

G = (C₂xC₄).SD₁₆ order 128 = 2⁷

7^th non-split extension by C₂xC₄ of SD₁₆ acting via SD₁₆/C₂=D₄