C16:D4 - GroupNames

G = C₁₆⋊D₄ order 128 = 2⁷

1^st semidirect product of C₁₆ and D₄ acting via D₄/C₂=C₂²

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

Aliases: C₁₆⋊₁D₄, C₂³.₁₆D₈, (C₂×D₁₆)⋊₈C₂, C₁₆⋊₄C₄⋊₄C₂, (C₂×C₄).₃₆D₈, C₈⋊₇D₄⋊₃₃C₂, (C₂×C₈).₂₁₄D₄, C₈.₁₀₂(C₂×D₄), C₂.D₁₆⋊₁₇C₂, C₈.₄₇(C₄○D₄), C₄.₂₀(C₄○D₈), (C₂×M₅(2))⋊₁C₂, (C₂×D₈).₇C₂², C₂.₂₁(C₈⋊₇D₄), C₄.₉₃(C₄⋊D₄), C₂.D₈.₈C₂², (C₂×C₁₆).₂₃C₂², (C₂×C₈).₅₂₃C₂³, C₂².₁₀₉(C₂×D₈), (C₂²×C₄).₃₄₉D₄, C₂.₁₂(C₁₆⋊C₂²), (C₂²×C₈).₂₉₉C₂², (C₂×C₄).₇₉₁(C₂×D₄), SmallGroup(128,950)

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

Derived series

C₁ — C₂×C₈ — C₁₆⋊D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₂×M₅(2) — C₁₆⋊D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₁₆⋊D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₁₆⋊D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₁₆⋊D₄

Generators and relations for C₁₆⋊D₄
G = < a,b,c | a¹⁶=b⁴=c²=1, bab^-1=a⁷, cac=a^-1, cbc=b^-1 >

Subgroups: 264 in 85 conjugacy classes, 32 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₆, C₁₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, C₂²×C₄, C₂×D₄, D₄⋊C₄, C₂.D₈, C₂×C₁₆, M₅(2), D₁₆, C₄⋊D₄, C₂²×C₈, C₂×D₈, C₂.D₁₆, C₁₆⋊₄C₄, C₈⋊₇D₄, C₂×M₅(2), C₂×D₁₆, C₁₆⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₄○D₈, C₈⋊₇D₄, C₁₆⋊C₂², C₁₆⋊D₄

Character table of C₁₆⋊D₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	4	16	16	2	2	4	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	-2	0	0	2	2	-2	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	2	0	0	2	2	2	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	0	0	2	-2	0	0	0	-2	2	2	-2	0	0	-2	0	2	0	2	0	0	-2	orthogonal lifted from D₄
ρ₁₂	2	-2	2	-2	0	0	0	2	-2	0	0	0	-2	2	2	-2	0	0	2	0	-2	0	-2	0	0	2	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	0	0	-2	-2	2	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	2	0	0	-2	-2	-2	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	2	0	0	-2	-2	-2	0	0	0	0	0	0	0	0	-√2	-√2	-√2	-√2	√2	√2	√2	√2	orthogonal lifted from D₈
ρ₁₆	2	2	2	2	-2	0	0	-2	-2	2	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₇	2	-2	2	-2	0	0	0	2	-2	0	0	0	2	-2	-2	2	0	0	0	-2i	0	2i	0	2i	-2i	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	0	0	0	2	-2	0	0	0	2	-2	-2	2	0	0	0	2i	0	-2i	0	-2i	2i	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	0	-2i	2i	-√2	√-2	√2	-√-2	-√2	√-2	-√-2	√2	complex lifted from C₄○D₈
ρ₂₀	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	0	2i	-2i	-√2	-√-2	√2	√-2	-√2	-√-2	√-2	√2	complex lifted from C₄○D₈
ρ₂₁	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	0	2i	-2i	√2	√-2	-√2	-√-2	√2	√-2	-√-2	-√2	complex lifted from C₄○D₈
ρ₂₂	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	0	-2i	2i	√2	-√-2	-√2	√-2	√2	-√-2	√-2	-√2	complex lifted from C₄○D₈
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	orthogonal lifted from C₁₆⋊C₂²
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	orthogonal lifted from C₁₆⋊C₂²

Smallest permutation representation of C₁₆⋊D₄
►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 62 48 32)(2 53 33 23)(3 60 34 30)(4 51 35 21)(5 58 36 28)(6 49 37 19)(7 56 38 26)(8 63 39 17)(9 54 40 24)(10 61 41 31)(11 52 42 22)(12 59 43 29)(13 50 44 20)(14 57 45 27)(15 64 46 18)(16 55 47 25)

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

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

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

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

Matrix representation of C₁₆⋊D₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	9	4	10	6
0	0	0	11	16	8
0	0	6	7	11	10
0	0	11	1	7	3

16	2	0	0	0	0
16	1	0	0	0	0
0	0	6	7	11	10
0	0	6	2	13	15
0	0	9	4	10	6
0	0	6	7	9	16

16	0	0	0	0	0
16	1	0	0	0	0
0	0	9	4	10	6
0	0	0	6	1	9
0	0	6	7	11	10
0	0	5	1	0	8

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,9,0,6,11,0,0,4,11,7,1,0,0,10,16,11,7,0,0,6,8,10,3],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,6,6,9,6,0,0,7,2,4,7,0,0,11,13,10,9,0,0,10,15,6,16],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,9,0,6,5,0,0,4,6,7,1,0,0,10,1,11,0,0,0,6,9,10,8] >;

C₁₆⋊D₄ in GAP, Magma, Sage, TeX

C_{16}\rtimes D_4

% in TeX

G:=Group("C16:D4");

// GroupNames label

G:=SmallGroup(128,950);

// by ID

G=gap.SmallGroup(128,950);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,723,1123,360,4037,124]);

// Polycyclic

G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;

// generators/relations

Export

Character table of C₁₆⋊D₄ in TeX

G = C16⋊D4 order 128 = 27

1st semidirect product of C16 and D4 acting via D4/C2=C22

G = C₁₆⋊D₄ order 128 = 2⁷

1^st semidirect product of C₁₆ and D₄ acting via D₄/C₂=C₂²