C2^3.2SD16 - GroupNames

G = C₂³.₂SD₁₆ order 128 = 2⁷

2^nd non-split extension by C₂³ of SD₁₆ acting via SD₁₆/C₂=D₄

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₂³.₂SD₁₆, C₄.₈C₄≀C₂, (C₂×C₄).₂D₈, (C₂×C₈).₂₂D₄, C₄.Q₈.₂C₄, (C₂×Q₁₆).₄C₄, C₄.₄(C₂³⋊C₄), C₈.D₄.₂C₂, C₂³.C₈.₅C₂, (C₂²×C₄).₃₀D₄, C₄.₁₀C₄².₁C₂, C₂.₇(C₂².SD₁₆), (C₂×M₄(2)).₄C₂², C₂².₁₆(D₄⋊C₄), (C₂×C₈).₄(C₂×C₄), (C₂×C₄).₅₅(C₂²⋊C₄), SmallGroup(128,74)

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

Derived series

C₁ — C₂×C₈ — C₂³.₂SD₁₆

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×M₄(2) — C₈.D₄ — C₂³.₂SD₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₂³.₂SD₁₆

Upper central

C₁ — C₂ — C₂×C₄ — C₂×M₄(2) — C₂³.₂SD₁₆

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₂×C₄ — C₂×M₄(2) — C₂³.₂SD₁₆

Generators and relations for C₂³.₂SD₁₆
G = < a,b,c,d,e | a²=b²=c²=1, d⁸=e²=c, dad^-1=eae^-1=ab=ba, ac=ca, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=abcd³ >

C₁

C₂

₂C₂

₄C₂

C₄

C₂²

C₄

₂C₂²

₂C₄

₄C₂²

₈C₄

C₂×C₄

C₂³

C₂×C₄

₂C₈

₂C₂×C₄

₂C₈

₄C₈

₄C₂×C₄

₄Q₈

₄C₂×C₄

₄C₈

C₂²×C₄

C₂×C₈

₂C₂×C₈

₂C₂×Q₈

₂C₄⋊C₄

₂C₂×C₈

₄M₄(2)

₄C₁₆

₄Q₁₆

₄M₄(2)

₄C₄⋊C₄

₄C₂²⋊C₄

₄M₄(2)

C₄.Q₈

C₂×M₄(2)

C₂×Q₁₆

₂M₅(2)

₂C₂×M₄(2)

₂Q₈⋊C₄

₂C₂²⋊Q₈

C₈.D₄

C₄.₁₀C₄²

C₂³.C₈

C₂³.₂SD₁₆

Character table of C₂³.₂SD₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	8A	8B	8C	8D	8E	8F	8G	16A	16B	16C	16D
size	1	1	2	4	2	2	4	16	16	4	4	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	trivial
ρ₂	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	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	linear of order 2
ρ₅	1	1	1	-1	1	1	-1	1	-1	-1	-1	i	i	-i	1	-i	i	i	-i	-i	linear of order 4
ρ₆	1	1	1	-1	1	1	-1	1	-1	-1	-1	-i	-i	i	1	i	-i	-i	i	i	linear of order 4
ρ₇	1	1	1	-1	1	1	-1	-1	1	-1	-1	i	i	-i	1	-i	-i	-i	i	i	linear of order 4
ρ₈	1	1	1	-1	1	1	-1	-1	1	-1	-1	-i	-i	i	1	i	i	i	-i	-i	linear of order 4
ρ₉	2	2	2	-2	2	2	-2	0	0	2	2	0	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	2	0	0	-2	-2	0	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₃	2	2	-2	0	2	-2	0	0	0	2i	-2i	-1-i	1+i	-1+i	0	1-i	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₄	2	2	-2	0	2	-2	0	0	0	-2i	2i	1-i	-1+i	1+i	0	-1-i	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₅	2	2	2	2	-2	-2	-2	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	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₇	2	2	-2	0	2	-2	0	0	0	-2i	2i	-1+i	1-i	-1-i	0	1+i	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₈	2	2	-2	0	2	-2	0	0	0	2i	-2i	1+i	-1-i	1-i	0	-1+i	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₉	4	4	-4	0	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₀	8	-8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of C₂³.₂SD₁₆
►On 32 points

Generators in S₃₂

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

(1 9)(3 11)(5 13)(7 15)(18 26)(20 28)(22 30)(24 32)

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

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

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

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

Matrix representation of C₂³.₂SD₁₆ ►in GL₈(𝔽₁₇)

1	0	0	15	0	0	0	0
0	16	2	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	16	0	0	0	0
15	0	14	2	0	0	0	16
0	9	8	1	0	0	1	0
0	9	8	1	0	1	0	0
15	0	14	2	16	0	0	0

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
2	7	13	13	1	0	0	0
14	9	4	4	0	1	0	0
14	9	4	4	0	0	1	0
2	7	13	13	0	0	0	1

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
0	0	0	0	16	0	0	0
0	0	0	0	0	16	0	0
0	0	0	0	0	0	16	0
0	0	0	0	0	0	0	16

14	9	4	4	0	0	2	0
15	10	4	4	0	0	0	15
0	0	0	0	1	0	0	16
0	0	0	0	0	16	1	0
2	15	13	2	2	15	0	5
1	16	8	8	15	2	1	11
8	16	9	1	15	2	1	11
2	16	12	12	2	15	0	5

0	16	2	0	0	0	0	0
1	0	0	15	0	0	0	0
0	0	0	16	0	0	0	0
0	0	1	0	0	0	0	0
13	8	11	16	14	3	4	13
13	16	1	11	3	3	13	13
15	6	14	7	4	13	3	14
0	13	0	5	13	13	14	14

G:=sub<GL(8,GF(17))| [1,0,0,0,15,0,0,15,0,16,0,0,0,9,9,0,0,2,1,0,14,8,8,14,15,0,0,16,2,1,1,2,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0],[16,0,0,0,2,14,14,2,0,16,0,0,7,9,9,7,0,0,16,0,13,4,4,13,0,0,0,16,13,4,4,13,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[14,15,0,0,2,1,8,2,9,10,0,0,15,16,16,16,4,4,0,0,13,8,9,12,4,4,0,0,2,8,1,12,0,0,1,0,2,15,15,2,0,0,0,16,15,2,2,15,2,0,0,1,0,1,1,0,0,15,16,0,5,11,11,5],[0,1,0,0,13,13,15,0,16,0,0,0,8,16,6,13,2,0,0,1,11,1,14,0,0,15,16,0,16,11,7,5,0,0,0,0,14,3,4,13,0,0,0,0,3,3,13,13,0,0,0,0,4,13,3,14,0,0,0,0,13,13,14,14] >;

C₂³.₂SD₁₆ in GAP, Magma, Sage, TeX

C_2^3._2{\rm SD}_{16}

% in TeX

G:=Group("C2^3.2SD16");

// GroupNames label

G:=SmallGroup(128,74);

// by ID

G=gap.SmallGroup(128,74);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,521,248,1411,172,4037,2028,124]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂³.₂SD₁₆ in TeX
Character table of C₂³.₂SD₁₆ in TeX

1	0	0	15	0	0	0	0
0	16	2	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	16	0	0	0	0
15	0	14	2	0	0	0	16
0	9	8	1	0	0	1	0
0	9	8	1	0	1	0	0
15	0	14	2	16	0	0	0

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
2	7	13	13	1	0	0	0
14	9	4	4	0	1	0	0
14	9	4	4	0	0	1	0
2	7	13	13	0	0	0	1

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
0	0	0	0	16	0	0	0
0	0	0	0	0	16	0	0
0	0	0	0	0	0	16	0
0	0	0	0	0	0	0	16

14	9	4	4	0	0	2	0
15	10	4	4	0	0	0	15
0	0	0	0	1	0	0	16
0	0	0	0	0	16	1	0
2	15	13	2	2	15	0	5
1	16	8	8	15	2	1	11
8	16	9	1	15	2	1	11
2	16	12	12	2	15	0	5

0	16	2	0	0	0	0	0
1	0	0	15	0	0	0	0
0	0	0	16	0	0	0	0
0	0	1	0	0	0	0	0
13	8	11	16	14	3	4	13
13	16	1	11	3	3	13	13
15	6	14	7	4	13	3	14
0	13	0	5	13	13	14	14

1	0	0	15	0	0	0	0
0	16	2	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	16	0	0	0	0
15	0	14	2	0	0	0	16
0	9	8	1	0	0	1	0
0	9	8	1	0	1	0	0
15	0	14	2	16	0	0	0

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
2	7	13	13	1	0	0	0
14	9	4	4	0	1	0	0
14	9	4	4	0	0	1	0
2	7	13	13	0	0	0	1

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
0	0	0	0	16	0	0	0
0	0	0	0	0	16	0	0
0	0	0	0	0	0	16	0
0	0	0	0	0	0	0	16

14	9	4	4	0	0	2	0
15	10	4	4	0	0	0	15
0	0	0	0	1	0	0	16
0	0	0	0	0	16	1	0
2	15	13	2	2	15	0	5
1	16	8	8	15	2	1	11
8	16	9	1	15	2	1	11
2	16	12	12	2	15	0	5

0	16	2	0	0	0	0	0
1	0	0	15	0	0	0	0
0	0	0	16	0	0	0	0
0	0	1	0	0	0	0	0
13	8	11	16	14	3	4	13
13	16	1	11	3	3	13	13
15	6	14	7	4	13	3	14
0	13	0	5	13	13	14	14

G = C23.2SD16 order 128 = 27

2nd non-split extension by C23 of SD16 acting via SD16/C2=D4

G = C₂³.₂SD₁₆ order 128 = 2⁷

2^nd non-split extension by C₂³ of SD₁₆ acting via SD₁₆/C₂=D₄

1	0	0	15	0	0	0	0
0	16	2	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	16	0	0	0	0
15	0	14	2	0	0	0	16
0	9	8	1	0	0	1	0
0	9	8	1	0	1	0	0
15	0	14	2	16	0	0	0

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
2	7	13	13	1	0	0	0
14	9	4	4	0	1	0	0
14	9	4	4	0	0	1	0
2	7	13	13	0	0	0	1

16	0	0	0	0	0	0	0
0	16	0	0	0	0	0	0
0	0	16	0	0	0	0	0
0	0	0	16	0	0	0	0
0	0	0	0	16	0	0	0
0	0	0	0	0	16	0	0
0	0	0	0	0	0	16	0
0	0	0	0	0	0	0	16

14	9	4	4	0	0	2	0
15	10	4	4	0	0	0	15
0	0	0	0	1	0	0	16
0	0	0	0	0	16	1	0
2	15	13	2	2	15	0	5
1	16	8	8	15	2	1	11
8	16	9	1	15	2	1	11
2	16	12	12	2	15	0	5

0	16	2	0	0	0	0	0
1	0	0	15	0	0	0	0
0	0	0	16	0	0	0	0
0	0	1	0	0	0	0	0
13	8	11	16	14	3	4	13
13	16	1	11	3	3	13	13
15	6	14	7	4	13	3	14
0	13	0	5	13	13	14	14