C2^3.19D4 - GroupNames

G = C₂³.₁₉D₄ order 64 = 2⁶

12^nd non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂³.₁₉D₄, C₂.D₈⋊₇C₂, C₂²⋊C₈⋊₇C₂, C₄.Q₈⋊₁₀C₂, (C₂×C₄).₁₀₆D₄, C₄⋊D₄.₇C₂, D₄⋊C₄⋊₁₃C₂, C₄²⋊C₂⋊₅C₂, C₄.₃₁(C₄○D₄), C₂.₁₅(C₄○D₈), C₄⋊C₄.₆₅C₂², (C₂×C₈).₃₈C₂², C₂.₁₈(C₈⋊C₂²), (C₂×C₄).₁₀₇C₂³, (C₂×D₄).₂₃C₂², C₂².₁₀₃(C₂×D₄), (C₂²×C₄).₅₃C₂², C₂.₁₃(C₂².D₄), SmallGroup(64,163)

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

Derived series

C₁ — C₂×C₄ — C₂³.₁₉D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄⋊C₄ — C₄²⋊C₂ — C₂³.₁₉D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂³.₁₉D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂³.₁₉D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂³.₁₉D₄

Generators and relations for C₂³.₁₉D₄
G = < a,b,c,d,e | a²=b²=c²=e²=1, d⁴=c, dad^-1=ab=ba, ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bcd³ >

C₁

C₂

₄C₂

₈C₂

C₄

C₂²

₂C₂²

₂C₄

₄C₄

₄C₂²

₄C₄

C₂×C₄

C₂³

₂C₂×C₄

₂C₂³

₂C₂×C₄

₂C₈

₂D₄

₂C₂×C₄

₂D₄

₄D₄

C₄⋊C₄

C₂×C₈

C₄⋊C₄

C₂²×C₄

C₂×D₄

₂C₂×D₄

₂C₄²

₂C₂²⋊C₄

C₄⋊D₄

C₂.D₈

D₄⋊C₄

C₄.Q₈

C₄²⋊C₂

C₂²⋊C₈

C₂³.₁₉D₄

Character table of C₂³.₁₉D₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D
size	1	1	1	1	4	8	2	2	2	2	4	4	4	4	8	4	4	4	4
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	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	linear of order 2
ρ₇	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	linear of order 2
ρ₉	2	2	2	2	2	0	-2	-2	-2	-2	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	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	0	0	-2	2	0	0	0	2i	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₂	2	-2	2	-2	0	0	-2	2	0	0	0	-2i	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₃	2	-2	2	-2	0	0	2	-2	0	0	2i	0	0	-2i	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₄	2	-2	2	-2	0	0	2	-2	0	0	-2i	0	0	2i	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	0	0	0	0	2i	-2i	0	0	0	0	0	-√2	√2	√-2	-√-2	complex lifted from C₄○D₈
ρ₁₆	2	-2	-2	2	0	0	0	0	-2i	2i	0	0	0	0	0	-√2	√2	-√-2	√-2	complex lifted from C₄○D₈
ρ₁₇	2	-2	-2	2	0	0	0	0	2i	-2i	0	0	0	0	0	√2	-√2	-√-2	√-2	complex lifted from C₄○D₈
ρ₁₈	2	-2	-2	2	0	0	0	0	-2i	2i	0	0	0	0	0	√2	-√2	√-2	-√-2	complex lifted from C₄○D₈
ρ₁₉	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²

Smallest permutation representation of C₂³.₁₉D₄
►On 32 points

Generators in S₃₂

(1 30)(2 13)(3 32)(4 15)(5 26)(6 9)(7 28)(8 11)(10 21)(12 23)(14 17)(16 19)(18 25)(20 27)(22 29)(24 31)

(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)

(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)

(2 22)(3 7)(4 20)(6 18)(8 24)(9 11)(10 28)(12 26)(13 15)(14 32)(16 30)(17 21)(25 31)(27 29)

G:=sub<Sym(32)| (1,30)(2,13)(3,32)(4,15)(5,26)(6,9)(7,28)(8,11)(10,21)(12,23)(14,17)(16,19)(18,25)(20,27)(22,29)(24,31), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,28)(12,26)(13,15)(14,32)(16,30)(17,21)(25,31)(27,29)>;

G:=Group( (1,30)(2,13)(3,32)(4,15)(5,26)(6,9)(7,28)(8,11)(10,21)(12,23)(14,17)(16,19)(18,25)(20,27)(22,29)(24,31), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,28)(12,26)(13,15)(14,32)(16,30)(17,21)(25,31)(27,29) );

G=PermutationGroup([[(1,30),(2,13),(3,32),(4,15),(5,26),(6,9),(7,28),(8,11),(10,21),(12,23),(14,17),(16,19),(18,25),(20,27),(22,29),(24,31)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(2,22),(3,7),(4,20),(6,18),(8,24),(9,11),(10,28),(12,26),(13,15),(14,32),(16,30),(17,21),(25,31),(27,29)]])

C₂³.₁₉D₄ is a maximal subgroup of
C₂⁴.₁₁₅D₄ C₂⁴.₁₁₆D₄ C₂⁴.₁₁₇D₄ C₄².₂₂₉D₄ C₄².₂₃₃D₄ C₂⁴.₁₂₁D₄ C₂⁴.₁₂₄D₄ C₂⁴.₁₂₇D₄ C₂⁴.₁₃₀D₄ C₄².₂₈₇D₄ C₄².₂₉₂D₄
(C₂×D₄).D_2p: (C₂×D₄).₃₀₁D₄ (C₂×D₄).₃₀₃D₄ (C₂×D₄).₃₀₄D₄ C₄.2₊¹⁺⁴ C₄.₁₄2₊¹⁺⁴ C₄.₁₅2₊¹⁺⁴ C₄.₁₆2₊¹⁺⁴ C₄².₄₆₁C₂³ ...
C₄⋊C₄.D_2p: C₄².₃₅₂C₂³ C₄².₃₅₅C₂³ C₄².₃₅₆C₂³ C₄².₃₆₀C₂³ C₄².₄₂₃C₂³ C₄².₄₂₅C₂³ C₄².₄₂₆C₂³ C₄.Q₈⋊S₃ ...
C_2p.(C₄○D₈): C₄².₃₈₄D₄ C₄².₄₅₀D₄ C₄².₂₈₀D₄ C₄².₂₈₅D₄ C₂³.₁₈D₁₂ C₂³.₁₃D₂₀ C₂³.₁₃D₂₈ ...
C₂³.₁₉D₄ is a maximal quotient of
C₂⁴.₆₉D₄ C₂⁴.₇₄D₄ C₄.Q₈⋊₁₀C₄ C₂.D₈⋊₅C₄ C₄²⋊₈C₄⋊C₂ C₄⋊C₄.Q₈
C₄⋊C₄.D_2p: C₂⁴.₇₁D₄ D₄⋊C₄⋊C₄ C₄.₆₇(C₄×D₄) C₂⁴.₈₃D₄ C₂⁴.₈₄D₄ (C₂×C₈).₂₄Q₈ D₆⋊C₈⋊₁₁C₂ C₂₄⋊₁C₄⋊C₂ ...
(C₂×C₈).D_2p: C₂⁴.₈₈D₄ C₂⁴.₈₉D₄ (C₂×C₈).₁₆₈D₄ C₂³.₁₈D₁₂ C₂³.₁₃D₂₀ C₂³.₁₃D₂₈ ...

Matrix representation of C₂³.₁₉D₄ ►in GL₄(𝔽₁₇) generated by

1	15	0	0
0	16	0	0
0	0	13	8
0	0	13	4

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

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

13	0	0	0
13	4	0	0
0	0	0	11
0	0	3	11

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

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

C₂³.₁₉D₄ in GAP, Magma, Sage, TeX

C_2^3._{19}D_4

% in TeX

G:=Group("C2^3.19D4");

// GroupNames label

G:=SmallGroup(64,163);

// by ID

G=gap.SmallGroup(64,163);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,199,362,50,1444,376,88]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^4=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e=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=b*c*d^3>;

// generators/relations

Export

Subgroup lattice of C₂³.₁₉D₄ in TeX
Character table of C₂³.₁₉D₄ in TeX

G = C23.19D4 order 64 = 26

12nd non-split extension by C23 of D4 acting via D4/C2=C22

G = C₂³.₁₉D₄ order 64 = 2⁶

12^nd non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²