C8:14SD16 - GroupNames

G = C₈⋊₁₄SD₁₆ order 128 = 2⁷

2^nd semidirect product of C₈ and SD₁₆ acting via SD₁₆/Q₈=C₂

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

Aliases: Q₈⋊₁SD₁₆, C₈⋊₁₄SD₁₆, C₄².₂₁₇C₂³, (C₈×Q₈)⋊₁₉C₂, C₈⋊₂C₈⋊₂₂C₂, C₄⋊C₄.₁₉₂D₄, (C₂×C₈).₃₆₄D₄, Q₈⋊Q₈⋊₃₅C₂, C₄.₆₅(C₄○D₈), C₈⋊₅D₄.₁₀C₂, C₂.₉(C₈⋊₈D₄), C₄⋊SD₁₆.₆C₂, (C₂×Q₈).₁₄₆D₄, C₄.₇₈(C₂×SD₁₆), C₄.D₈.₇C₂, C₄⋊Q₈.₄₂C₂², C₄.₁₀D₈⋊₂₂C₂, C₄⋊C₈.₁₇₃C₂², (C₄×C₈).₂₅₀C₂², C₂.₆(C₄⋊SD₁₆), C₄.₁₁₅(C₈⋊C₂²), C₂.₉(D₄.₃D₄), C₄⋊₁D₄.₂₅C₂², (C₄×Q₈).₂₆₁C₂², C₂².₁₇₈(C₄⋊D₄), (C₂×C₄).₂(C₄○D₄), (C₂×C₄).₁₂₅₂(C₂×D₄), SmallGroup(128,398)

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

Derived series

C₁ — C₄² — C₈⋊₁₄SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈×Q₈ — C₈⋊₁₄SD₁₆

Lower central

C₁ — C₂² — C₄² — C₈⋊₁₄SD₁₆

Upper central

C₁ — C₂² — C₄² — C₈⋊₁₄SD₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — C₈⋊₁₄SD₁₆

Generators and relations for C₈⋊₁₄SD₁₆
G = < a,b,c | a⁸=b⁸=c²=1, bab^-1=cac=a³, cbc=b³ >

Subgroups: 216 in 86 conjugacy classes, 36 normal (32 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₄×C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₄×Q₈, C₄⋊₁D₄, C₄⋊Q₈, C₂×SD₁₆, C₄.D₈, C₄.₁₀D₈, C₈⋊₂C₈, C₈×Q₈, C₄⋊SD₁₆, Q₈⋊Q₈, C₈⋊₅D₄, C₈⋊₁₄SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₄○D₈, C₈⋊C₂², C₄⋊SD₁₆, C₈⋊₈D₄, D₄.₃D₄, C₈⋊₁₄SD₁₆

Character table of C₈⋊₁₄SD₁₆

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	16	2	2	2	2	4	4	4	4	4	16	2	2	2	2	4	4	4	4	4	4	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	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	-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	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	-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	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	-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	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	-1	1	-1	linear of order 2
ρ₉	2	2	2	2	0	-2	2	-2	2	0	0	-2	0	0	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	-2	2	-2	2	0	0	-2	0	0	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	2	-2	2	-2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	2	-2	2	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	-2	-2	-2	-2	0	0	2	0	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₄	2	2	2	2	0	-2	-2	-2	-2	0	0	2	0	0	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	2	-2	0	0	2	0	-2	0	0	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₁₆	2	2	-2	-2	0	-2	0	2	0	2i	0	0	-2i	0	0	-√-2	√-2	-√-2	√-2	-√2	√2	-√2	-√-2	√-2	√2	0	0	0	0	complex lifted from C₄○D₈
ρ₁₇	2	-2	2	-2	0	0	2	0	-2	0	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₈	2	2	-2	-2	0	2	0	-2	0	0	2	0	0	-2	0	-√-2	√-2	-√-2	√-2	√-2	-√-2	-√-2	√-2	-√-2	√-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₉	2	2	-2	-2	0	2	0	-2	0	0	-2	0	0	2	0	√-2	-√-2	√-2	-√-2	√-2	-√-2	-√-2	-√-2	√-2	√-2	0	0	0	0	complex lifted from SD₁₆
ρ₂₀	2	2	-2	-2	0	-2	0	2	0	-2i	0	0	2i	0	0	√-2	-√-2	√-2	-√-2	-√2	√2	-√2	√-2	-√-2	√2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₁	2	2	-2	-2	0	-2	0	2	0	-2i	0	0	2i	0	0	-√-2	√-2	-√-2	√-2	√2	-√2	√2	-√-2	√-2	-√2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₂	2	2	-2	-2	0	-2	0	2	0	2i	0	0	-2i	0	0	√-2	-√-2	√-2	-√-2	√2	-√2	√2	√-2	-√-2	-√2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₃	2	2	-2	-2	0	2	0	-2	0	0	2	0	0	-2	0	√-2	-√-2	√-2	-√-2	-√-2	√-2	√-2	-√-2	√-2	-√-2	0	0	0	0	complex lifted from SD₁₆
ρ₂₄	2	-2	2	-2	0	0	2	0	-2	0	0	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₂₅	2	2	-2	-2	0	2	0	-2	0	0	-2	0	0	2	0	-√-2	√-2	-√-2	√-2	-√-2	√-2	√-2	√-2	-√-2	-√-2	0	0	0	0	complex lifted from SD₁₆
ρ₂₆	2	-2	2	-2	0	0	2	0	-2	0	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₇	4	-4	4	-4	0	0	-4	0	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
ρ₂₈	4	-4	-4	4	0	0	0	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	complex lifted from D₄.₃D₄
ρ₂₉	4	-4	-4	4	0	0	0	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	complex lifted from D₄.₃D₄

Smallest permutation representation of C₈⋊₁₄SD₁₆
►On 64 points

Generators in S₆₄

(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)

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

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

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

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

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

Matrix representation of C₈⋊₁₄SD₁₆ ►in GL₄(𝔽₁₇) generated by

12	5	0	0
12	12	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	16	0	0
0	0	5	5
0	0	12	5

1	0	0	0
0	16	0	0
0	0	1	0
0	0	0	16

G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,5,12,0,0,5,5],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C₈⋊₁₄SD₁₆ in GAP, Magma, Sage, TeX

C_8\rtimes_{14}{\rm SD}_{16}

% in TeX

G:=Group("C8:14SD16");

// GroupNames label

G:=SmallGroup(128,398);

// by ID

G=gap.SmallGroup(128,398);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,352,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₁₄SD₁₆ in TeX

G = C8⋊14SD16 order 128 = 27

2nd semidirect product of C8 and SD16 acting via SD16/Q8=C2

G = C₈⋊₁₄SD₁₆ order 128 = 2⁷

2^nd semidirect product of C₈ and SD₁₆ acting via SD₁₆/Q₈=C₂