C8.12SD16 - GroupNames

G = C₈.₁₂SD₁₆ order 128 = 2⁷

12^nd non-split extension by C₈ of SD₁₆ acting via SD₁₆/C₄=C₂²

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

Aliases: C₈.₁₂SD₁₆, C₄².₁₅₃D₄, (C₂×C₄).₄₅D₈, C₈⋊₂Q₈⋊₂₆C₂, C₁₆⋊₅C₄⋊₁₂C₂, (C₂×C₈).₁₃₅D₄, C₂.D₁₆.₇C₂, C₈.₅₃(C₄○D₄), C₄.₁₇(C₂×SD₁₆), C₂.Q₃₂⋊₁₉C₂, (C₄×C₈).₁₆₇C₂², (C₂×C₈).₅₄₁C₂³, (C₂×C₁₆).₅₈C₂², C₈.₁₂D₄.₅C₂, (C₂×D₈).₁₄C₂², C₂².₁₂₇(C₂×D₈), C₂.₂₀(C₁₆⋊C₂²), C₂.D₈.₂₆C₂², C₂.₂₀(Q₃₂⋊C₂), C₄.₁₀(C₄.₄D₄), C₂.₁₂(C₄.₄D₈), (C₂×Q₁₆).₁₅C₂², (C₂×C₄).₈₀₉(C₂×D₄), SmallGroup(128,975)

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

Derived series

C₁ — C₂×C₈ — C₈.₁₂SD₁₆

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×C₁₆ — C₁₆⋊₅C₄ — C₈.₁₂SD₁₆

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₈.₁₂SD₁₆

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — C₈.₁₂SD₁₆

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₈.₁₂SD₁₆

Generators and relations for C₈.₁₂SD₁₆
G = < a,b,c | a⁸=c²=1, b⁸=a⁴, bab^-1=a⁵, cac=a³, cbc=a⁶b³ >

Subgroups: 200 in 71 conjugacy classes, 32 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₁₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄×C₈, C₂.D₈, C₂.D₈, C₂×C₁₆, C₄.₄D₄, C₄⋊Q₈, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₁₆⋊₅C₄, C₂.D₁₆, C₂.Q₃₂, C₈.₁₂D₄, C₈⋊₂Q₈, C₈.₁₂SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, SD₁₆, C₂×D₄, C₄○D₄, C₄.₄D₄, C₂×D₈, C₂×SD₁₆, C₄.₄D₈, C₁₆⋊C₂², Q₃₂⋊C₂, C₈.₁₂SD₁₆

Character table of C₈.₁₂SD₁₆

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	16	2	2	4	4	16	16	16	2	2	2	2	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	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	2	2	2	2	0	2	2	-2	-2	0	0	0	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	2	2	2	2	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₂	2	2	2	2	0	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₃	2	2	2	2	0	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	2	0	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	2	-2	0	2	-2	0	0	0	0	0	2	-2	2	-2	0	0	2i	0	-2i	0	2i	-2i	0	0	complex lifted from C₄○D₄
ρ₁₆	2	-2	2	-2	0	-2	2	0	0	0	0	0	0	0	0	0	2	-2	-√-2	√-2	-√-2	-√-2	√-2	√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₇	2	-2	2	-2	0	2	-2	0	0	0	0	0	-2	2	-2	2	0	0	0	2i	0	2i	0	0	-2i	-2i	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	0	2	-2	0	0	0	0	0	-2	2	-2	2	0	0	0	-2i	0	-2i	0	0	2i	2i	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	0	2	-2	0	0	0	0	0	2	-2	2	-2	0	0	-2i	0	2i	0	-2i	2i	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	2	-2	0	-2	2	0	0	0	0	0	0	0	0	0	-2	2	-√-2	-√-2	-√-2	√-2	√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₁	2	-2	2	-2	0	-2	2	0	0	0	0	0	0	0	0	0	-2	2	√-2	√-2	√-2	-√-2	-√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₂₂	2	-2	2	-2	0	-2	2	0	0	0	0	0	0	0	0	0	2	-2	√-2	-√-2	√-2	√-2	-√-2	-√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	0	0	-2√2	2√2	2√2	-2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²
ρ₂₄	4	-4	-4	4	0	0	0	0	0	0	0	0	2√2	-2√2	-2√2	2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²
ρ₂₅	4	4	-4	-4	0	0	0	0	0	0	0	0	-2√2	-2√2	2√2	2√2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₃₂⋊C₂, Schur index 2
ρ₂₆	4	4	-4	-4	0	0	0	0	0	0	0	0	2√2	2√2	-2√2	-2√2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₃₂⋊C₂, Schur index 2

Smallest permutation representation of C₈.₁₂SD₁₆
►On 64 points

Generators in S₆₄

(1 38 28 59 9 46 20 51)(2 47 29 52 10 39 21 60)(3 40 30 61 11 48 22 53)(4 33 31 54 12 41 23 62)(5 42 32 63 13 34 24 55)(6 35 17 56 14 43 25 64)(7 44 18 49 15 36 26 57)(8 37 19 58 16 45 27 50)

(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)

(2 23)(3 15)(4 21)(5 13)(6 19)(7 11)(8 17)(10 31)(12 29)(14 27)(16 25)(18 30)(20 28)(22 26)(33 47)(34 63)(35 45)(36 61)(37 43)(38 59)(39 41)(40 57)(42 55)(44 53)(46 51)(48 49)(50 64)(52 62)(54 60)(56 58)

G:=sub<Sym(64)| (1,38,28,59,9,46,20,51)(2,47,29,52,10,39,21,60)(3,40,30,61,11,48,22,53)(4,33,31,54,12,41,23,62)(5,42,32,63,13,34,24,55)(6,35,17,56,14,43,25,64)(7,44,18,49,15,36,26,57)(8,37,19,58,16,45,27,50), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (2,23)(3,15)(4,21)(5,13)(6,19)(7,11)(8,17)(10,31)(12,29)(14,27)(16,25)(18,30)(20,28)(22,26)(33,47)(34,63)(35,45)(36,61)(37,43)(38,59)(39,41)(40,57)(42,55)(44,53)(46,51)(48,49)(50,64)(52,62)(54,60)(56,58)>;

G:=Group( (1,38,28,59,9,46,20,51)(2,47,29,52,10,39,21,60)(3,40,30,61,11,48,22,53)(4,33,31,54,12,41,23,62)(5,42,32,63,13,34,24,55)(6,35,17,56,14,43,25,64)(7,44,18,49,15,36,26,57)(8,37,19,58,16,45,27,50), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (2,23)(3,15)(4,21)(5,13)(6,19)(7,11)(8,17)(10,31)(12,29)(14,27)(16,25)(18,30)(20,28)(22,26)(33,47)(34,63)(35,45)(36,61)(37,43)(38,59)(39,41)(40,57)(42,55)(44,53)(46,51)(48,49)(50,64)(52,62)(54,60)(56,58) );

G=PermutationGroup([[(1,38,28,59,9,46,20,51),(2,47,29,52,10,39,21,60),(3,40,30,61,11,48,22,53),(4,33,31,54,12,41,23,62),(5,42,32,63,13,34,24,55),(6,35,17,56,14,43,25,64),(7,44,18,49,15,36,26,57),(8,37,19,58,16,45,27,50)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(2,23),(3,15),(4,21),(5,13),(6,19),(7,11),(8,17),(10,31),(12,29),(14,27),(16,25),(18,30),(20,28),(22,26),(33,47),(34,63),(35,45),(36,61),(37,43),(38,59),(39,41),(40,57),(42,55),(44,53),(46,51),(48,49),(50,64),(52,62),(54,60),(56,58)]])

Matrix representation of C₈.₁₂SD₁₆ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	7	1	12	8
0	0	1	9	15	7
0	0	10	6	5	13
0	0	15	2	12	13

12	5	0	0	0	0
12	12	0	0	0	0
0	0	16	6	9	8
0	0	10	3	8	0
0	0	12	14	7	7
0	0	15	9	0	8

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	16	16	0	0
0	0	9	15	14	14
0	0	6	14	14	3

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

C₈.₁₂SD₁₆ in GAP, Magma, Sage, TeX

C_8._{12}{\rm SD}_{16}

% in TeX

G:=Group("C8.12SD16");

// GroupNames label

G:=SmallGroup(128,975);

// by ID

G=gap.SmallGroup(128,975);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,120,422,723,58,1123,360,3924,102,4037,124]);

// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=a^6*b^3>;

// generators/relations

Export

Character table of C₈.₁₂SD₁₆ in TeX

G = C8.12SD16 order 128 = 27

12nd non-split extension by C8 of SD16 acting via SD16/C4=C22

G = C₈.₁₂SD₁₆ order 128 = 2⁷

12^nd non-split extension by C₈ of SD₁₆ acting via SD₁₆/C₄=C₂²