C2^3.13SD16 - GroupNames

G = C₂³.₁₃SD₁₆ order 128 = 2⁷

3^rd non-split extension by C₂³ of SD₁₆ acting via SD₁₆/C₄=C₂²

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

Aliases: C₂³.₁₃SD₁₆, C₄.₂₁C₄≀C₂, C₂.D₈.₂C₄, (C₂×C₄).₁₀₃D₈, (C₂×C₈).₃₀₃D₄, (C₂×Q₁₆).₁C₄, C₂²⋊C₁₆.₄C₂, C₄.₈(C₂³⋊C₄), C₈.₁₈D₄.₂C₂, (C₂²×C₄).₁₈₈D₄, C₂.₅(D₈.C₄), C₂.₃(C₈.₁₇D₄), C₄.C₄².₆C₂, (C₂²×C₈).₁₀₀C₂², C₂².₅₈(D₄⋊C₄), C₂.₁₅(C₂².SD₁₆), (C₂×C₈).₂₀(C₂×C₄), (C₂×C₄).₂₂₀(C₂²⋊C₄), SmallGroup(128,82)

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

Derived series

C₁ — C₂×C₈ — C₂³.₁₃SD₁₆

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₈.₁₈D₄ — C₂³.₁₃SD₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₂³.₁₃SD₁₆

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₁₃SD₁₆

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₂×C₄ — C₂²×C₈ — C₂³.₁₃SD₁₆

Generators and relations for C₂³.₁₃SD₁₆
G = < a,b,c,d,e | a²=b²=c²=1, d⁸=e²=c, eae^-1=ab=ba, ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=abd³ >

C₁

C₂

₄C₂

C₄

C₂²

C₄

₂C₂²

₂C₄

₈C₄

C₂×C₄

C₂³

₂C₂×C₄

₂C₈

₂C₂×C₄

₄C₈

₄Q₈

₄C₂×C₄

₄Q₈

₄C₈

C₂×C₈

C₂²×C₄

C₂×C₈

₂M₄(2)

₂C₄⋊C₄

₂C₂×Q₈

₂M₄(2)

₄C₁₆

₄C₂×C₈

₄C₄⋊C₄

₄Q₁₆

₄C₂×C₈

₄M₄(2)

₄C₂²⋊C₄

C₂²×C₈

C₂.D₈

C₂×Q₁₆

₂C₂×C₁₆

₂Q₈⋊C₄

₂C₂²⋊Q₈

₂C₂×M₄(2)

C₄.C₄²

C₂²⋊C₁₆

C₈.₁₈D₄

C₂³.₁₃SD₁₆

Character table of C₂³.₁₃SD₁₆

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	4	2	2	2	2	16	16	2	2	2	2	4	4	8	8	8	8	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	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	-i	-i	i	i	i	-i	i	i	-i	-i	-i	i	linear of order 4
ρ₆	1	1	1	1	-1	-1	1	1	-1	1	-1	-1	-1	-1	-1	1	1	-i	-i	i	i	-i	i	-i	-i	i	i	i	-i	linear of order 4
ρ₇	1	1	1	1	-1	-1	1	1	-1	1	-1	-1	-1	-1	-1	1	1	i	i	-i	-i	i	-i	i	i	-i	-i	-i	i	linear of order 4
ρ₈	1	1	1	1	-1	-1	1	1	-1	-1	1	-1	-1	-1	-1	1	1	i	i	-i	-i	-i	i	-i	-i	i	i	i	-i	linear of order 4
ρ₉	2	2	2	2	-2	-2	2	2	-2	0	0	2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	2	2	2	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	2	-2	-2	2	0	0	0	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	-2	2	-2	-2	2	0	0	0	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	0	2	-2	0	0	0	-2i	2i	2i	-2i	0	0	-1+i	1-i	-1-i	1+i	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₄	2	-2	2	-2	0	0	2	-2	0	0	0	2i	-2i	-2i	2i	0	0	-1-i	1+i	-1+i	1-i	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₅	2	-2	2	-2	0	0	2	-2	0	0	0	-2i	2i	2i	-2i	0	0	1-i	-1+i	1+i	-1-i	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₆	2	-2	2	-2	0	0	2	-2	0	0	0	2i	-2i	-2i	2i	0	0	1+i	-1-i	1-i	-1+i	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₇	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	√-2	-√-2	√-2	-√-2	√-2	-√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₈	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	-√-2	√-2	-√-2	√-2	-√-2	√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₉	2	-2	-2	2	0	2i	0	0	-2i	0	0	√-2	-√-2	√-2	-√-2	-√2	√2	0	0	0	0	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆³+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁵	complex lifted from D₈.C₄
ρ₂₀	2	-2	-2	2	0	-2i	0	0	2i	0	0	-√-2	√-2	-√-2	√-2	-√2	√2	0	0	0	0	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆	complex lifted from D₈.C₄
ρ₂₁	2	-2	-2	2	0	-2i	0	0	2i	0	0	√-2	-√-2	√-2	-√-2	√2	-√2	0	0	0	0	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆³+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	complex lifted from D₈.C₄
ρ₂₂	2	-2	-2	2	0	2i	0	0	-2i	0	0	-√-2	√-2	-√-2	√-2	√2	-√2	0	0	0	0	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆³+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆⁹	complex lifted from D₈.C₄
ρ₂₃	2	-2	-2	2	0	-2i	0	0	2i	0	0	-√-2	√-2	-√-2	√-2	-√2	√2	0	0	0	0	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	complex lifted from D₈.C₄
ρ₂₄	2	-2	-2	2	0	2i	0	0	-2i	0	0	-√-2	√-2	-√-2	√-2	√2	-√2	0	0	0	0	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆³+ζ₁₆	complex lifted from D₈.C₄
ρ₂₅	2	-2	-2	2	0	2i	0	0	-2i	0	0	√-2	-√-2	√-2	-√-2	-√2	√2	0	0	0	0	ζ₁₆³+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	complex lifted from D₈.C₄
ρ₂₆	2	-2	-2	2	0	-2i	0	0	2i	0	0	√-2	-√-2	√-2	-√-2	√2	-√2	0	0	0	0	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆⁹+ζ₁₆³	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆¹³	complex lifted from D₈.C₄
ρ₂₇	4	-4	4	-4	0	0	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₈	4	4	-4	-4	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	0	0	0	0	symplectic lifted from C₈.₁₇D₄, Schur index 2
ρ₂₉	4	4	-4	-4	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	0	0	0	0	symplectic lifted from C₈.₁₇D₄, Schur index 2

Smallest permutation representation of C₂³.₁₃SD₁₆
►On 64 points

Generators in S₆₄

(1 54)(3 56)(5 58)(7 60)(9 62)(11 64)(13 50)(15 52)(17 35)(18 26)(19 37)(20 28)(21 39)(22 30)(23 41)(24 32)(25 43)(27 45)(29 47)(31 33)(34 42)(36 44)(38 46)(40 48)

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

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

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

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

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

Matrix representation of C₂³.₁₃SD₁₆ ►in GL₄(𝔽₁₇) generated by

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

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

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

6	0	0	0
0	5	0	0
0	0	3	9
0	0	8	14

0	1	0	0
16	0	0	0
0	0	5	15
0	0	12	12

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

C₂³.₁₃SD₁₆ in GAP, Magma, Sage, TeX

C_2^3._{13}{\rm SD}_{16}

% in TeX

G:=Group("C2^3.13SD16");

// GroupNames label

G:=SmallGroup(128,82);

// by ID

G=gap.SmallGroup(128,82);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,248,2804,1411,172,4037,2028,124]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂³.₁₃SD₁₆ in TeX
Character table of C₂³.₁₃SD₁₆ in TeX

G = C23.13SD16 order 128 = 27

3rd non-split extension by C23 of SD16 acting via SD16/C4=C22

G = C₂³.₁₃SD₁₆ order 128 = 2⁷

3^rd non-split extension by C₂³ of SD₁₆ acting via SD₁₆/C₄=C₂²