C2^3.12SD16 - GroupNames

G = C₂³.₁₂SD₁₆ order 128 = 2⁷

2^nd non-split extension by C₂³ of SD₁₆ acting via SD₁₆/C₄=C₂²

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

Aliases: C₂³.₁₂SD₁₆, C₄.₂₀C₄≀C₂, (C₂×D₈).₁C₄, C₂.D₈.₁C₄, C₂²⋊C₁₆⋊₄C₂, (C₂×C₈).₃₀₂D₄, C₈⋊₇D₄.₂C₂, (C₂×C₄).₁₀₂D₈, C₄.₇(C₂³⋊C₄), (C₂²×C₄).₁₈₇D₄, C₄.C₄²⋊₁₁C₂, C₂.₄(D₈.C₄), C₂.₃(M₅(2)⋊C₂), (C₂²×C₈).₉₉C₂², C₂².₅₇(D₄⋊C₄), C₂.₁₄(C₂².SD₁₆), (C₂×C₈).₁₉(C₂×C₄), (C₂×C₄).₂₁₉(C₂²⋊C₄), SmallGroup(128,81)

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

Derived series

C₁ — C₂×C₈ — C₂³.₁₂SD₁₆

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₈⋊₇D₄ — C₂³.₁₂SD₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₂³.₁₂SD₁₆

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₁₂SD₁₆

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₂×C₄ — C₂²×C₈ — C₂³.₁₂SD₁₆

Generators and relations for C₂³.₁₂SD₁₆
G = < a,b,c,d,e | a²=b²=c²=e²=1, d⁸=c, eae=ab=ba, ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=abcd³ >

C₁

C₂

₄C₂

₁₆C₂

C₄

C₂²

C₄

₂C₂²

₂C₄

₂C₂²

₈C₂²

₈C₄

₈C₂²

C₂×C₄

C₂³

₂C₂×C₄

₂C₈

₂C₂×C₄

₂C₈

₄C₂³

₄D₄

₄C₈

₄C₂×C₄

₄C₈

₈D₄

C₂×C₈

C₂²×C₄

C₂×C₈

₂M₄(2)

₂C₂×D₄

₂C₄⋊C₄

₂M₄(2)

₄C₁₆

₄C₂²⋊C₄

₄M₄(2)

₄D₈

₄C₂×C₈

₄C₂×D₄

₄C₂×C₈

C₂²×C₈

C₂×D₈

C₂.D₈

₂C₂×C₁₆

₂D₄⋊C₄

₂C₄⋊D₄

₂C₂×M₄(2)

C₄.C₄²

C₂²⋊C₁₆

C₈⋊₇D₄

C₂³.₁₂SD₁₆

Character table of C₂³.₁₂SD₁₆

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	4	16	2	2	2	2	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	0	-2	2	2	-2	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	0	2	2	2	2	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	0	2	-2	-2	2	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	0	2	-2	-2	2	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	0	-2	2	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	0	-2	2	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	0	-2	2	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	0	-2	2	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	0	-2	-2	-2	-2	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	0	-2	-2	-2	-2	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	0	-2i	0	0	2i	0	√-2	-√-2	√-2	-√-2	√2	-√2	0	0	0	0	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆³+ζ₁₆	complex lifted from D₈.C₄
ρ₂₀	2	-2	-2	2	0	0	-2i	0	0	2i	0	√-2	-√-2	√-2	-√-2	√2	-√2	0	0	0	0	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	complex lifted from D₈.C₄
ρ₂₁	2	-2	-2	2	0	0	-2i	0	0	2i	0	-√-2	√-2	-√-2	√-2	-√2	√2	0	0	0	0	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹³+ζ₁₆⁷	complex lifted from D₈.C₄
ρ₂₂	2	-2	-2	2	0	0	2i	0	0	-2i	0	√-2	-√-2	√-2	-√-2	-√2	√2	0	0	0	0	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆³+ζ₁₆	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆⁹+ζ₁₆³	complex lifted from D₈.C₄
ρ₂₃	2	-2	-2	2	0	0	2i	0	0	-2i	0	√-2	-√-2	√-2	-√-2	-√2	√2	0	0	0	0	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	complex lifted from D₈.C₄
ρ₂₄	2	-2	-2	2	0	0	2i	0	0	-2i	0	-√-2	√-2	-√-2	√-2	√2	-√2	0	0	0	0	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆⁹+ζ₁₆³	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	complex lifted from D₈.C₄
ρ₂₅	2	-2	-2	2	0	0	2i	0	0	-2i	0	-√-2	√-2	-√-2	√-2	√2	-√2	0	0	0	0	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆³+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆⁵	complex lifted from D₈.C₄
ρ₂₆	2	-2	-2	2	0	0	-2i	0	0	2i	0	-√-2	√-2	-√-2	√-2	-√2	√2	0	0	0	0	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆³+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁵	complex lifted from D₈.C₄
ρ₂₇	4	-4	4	-4	0	0	0	4	-4	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	orthogonal lifted from M₅(2)⋊C₂
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	orthogonal lifted from M₅(2)⋊C₂

Smallest permutation representation of C₂³.₁₂SD₁₆
►On 64 points

Generators in S₆₄

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

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

(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)

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

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

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

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

Matrix representation of C₂³.₁₂SD₁₆ ►in GL₄(𝔽₁₇) generated by

13	8	0	0
13	4	0	0
0	0	16	16
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

5	11	0	0
3	16	0	0
0	0	16	6
0	0	2	1

1	0	0	0
1	16	0	0
0	0	1	0
0	0	15	16

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

C₂³.₁₂SD₁₆ in GAP, Magma, Sage, TeX

C_2^3._{12}{\rm SD}_{16}

% in TeX

G:=Group("C2^3.12SD16");

// GroupNames label

G:=SmallGroup(128,81);

// by ID

G=gap.SmallGroup(128,81);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂³.₁₂SD₁₆ in TeX
Character table of C₂³.₁₂SD₁₆ in TeX

G = C23.12SD16 order 128 = 27

2nd non-split extension by C23 of SD16 acting via SD16/C4=C22

G = C₂³.₁₂SD₁₆ order 128 = 2⁷

2^nd non-split extension by C₂³ of SD₁₆ acting via SD₁₆/C₄=C₂²