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 21)(3 23)(5 25)(7 27)(9 29)(11 31)(13 17)(15 19)(33 41)(34 50)(35 43)(36 52)(37 45)(38 54)(39 47)(40 56)(42 58)(44 60)(46 62)(48 64)(49 57)(51 59)(53 61)(55 63)

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

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

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

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

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

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