C2^3.38D4 - GroupNames

G = C₂³.₃₈D₄ order 64 = 2⁶

9^th non-split extension by C₂³ of D₄ acting via D₄/C₂²=C₂

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

Aliases: C₂³.₃₈D₄, (C₂×Q₈)⋊₆C₄, C₄.₅₂(C₂×D₄), Q₈.₆(C₂×C₄), (C₂×C₄).₁₂₅D₄, C₄.₆(C₂²×C₄), Q₈⋊C₄⋊₁₅C₂, C₄⋊C₄.₄₅C₂², (C₂×C₄).₆₄C₂³, (C₂×C₈).₄₄C₂², C₂².₄₆(C₂×D₄), (C₂²×Q₈).₅C₂, C₄.₁₅(C₂²⋊C₄), C₄²⋊C₂.₄C₂, C₂.₂(C₈.C₂²), (C₂×Q₈).₄₃C₂², (C₂×M₄(2)).₁₂C₂, (C₂²×C₄).₃₆C₂², C₂².₁₈(C₂²⋊C₄), (C₂×C₄).₂₃(C₂×C₄), C₂.₂₂(C₂×C₂²⋊C₄), SmallGroup(64,100)

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

Derived series

C₁ — C₄ — C₂³.₃₈D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂³.₃₈D₄

Lower central

C₁ — C₂ — C₄ — C₂³.₃₈D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂³.₃₈D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂³.₃₈D₄

Generators and relations for C₂³.₃₈D₄
G = < a,b,c,d,e | a²=b²=c²=1, d⁴=c, e²=b, ab=ba, dad^-1=eae^-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bd³ >

Subgroups: 113 in 75 conjugacy classes, 41 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, Q₈⋊C₄, C₄²⋊C₂, C₂×M₄(2), C₂²×Q₈, C₂³.₃₈D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₂²⋊C₄, C₈.C₂², C₂³.₃₈D₄

Character table of C₂³.₃₈D₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	8A	8B	8C	8D
size	1	1	1	1	2	2	2	2	2	2	4	4	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	trivial
ρ₂	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	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	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	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	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	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	linear of order 2
ρ₉	1	-1	1	-1	-1	1	1	-1	-1	1	i	i	1	1	-i	-i	-1	-1	i	i	-i	-i	linear of order 4
ρ₁₀	1	-1	1	-1	1	-1	1	-1	1	-1	-i	i	1	-1	i	-i	-1	1	-i	i	i	-i	linear of order 4
ρ₁₁	1	-1	1	-1	-1	1	1	-1	-1	1	-i	-i	-1	-1	i	i	1	1	i	i	-i	-i	linear of order 4
ρ₁₂	1	-1	1	-1	1	-1	1	-1	1	-1	i	-i	-1	1	-i	i	1	-1	-i	i	i	-i	linear of order 4
ρ₁₃	1	-1	1	-1	-1	1	1	-1	-1	1	i	i	-1	-1	-i	-i	1	1	-i	-i	i	i	linear of order 4
ρ₁₄	1	-1	1	-1	1	-1	1	-1	1	-1	-i	i	-1	1	i	-i	1	-1	i	-i	-i	i	linear of order 4
ρ₁₅	1	-1	1	-1	-1	1	1	-1	-1	1	-i	-i	1	1	i	i	-1	-1	-i	-i	i	i	linear of order 4
ρ₁₆	1	-1	1	-1	1	-1	1	-1	1	-1	i	-i	1	-1	-i	i	-1	1	i	-i	-i	i	linear of order 4
ρ₁₇	2	-2	2	-2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	2	2	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₂	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of C₂³.₃₈D₄
►On 32 points

Generators in S₃₂

(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)

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

(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)

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

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

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

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

C₂³.₃₈D₄ is a maximal subgroup of
(C₂²×Q₈)⋊C₄ C₄².₆D₄ M₄(2).₉D₄ C₂⁴.₉₈D₄ 2_-¹⁺⁴⋊₄C₄ C₄×C₈.C₂² C₂⁴.₁₇₈D₄ C₂⁴.₁₀₄D₄ (C₂×Q₈)⋊₁₆D₄ Q₈.(C₂×D₄) C₄².₄₄₅D₄ C₄².₄₄₆D₄ C₄².₂₂₀D₄ C₂⁴.₁₁₈D₄ (C₂×D₄).₃₀₂D₄ C₄².₂₄₁D₄ C₄².₂₄₂D₄ C₂³.₁₅S₄ (C₂×Q₈)⋊₄F₅
C₄⋊C₄.D_2p: C₈.C₂²⋊C₄ C₂⁴.₂₃D₄ C₄⋊Q₈⋊₁₅C₄ C₄.₁₀D₄⋊₃C₄ M₄(2).₃₁D₄ M₄(2).₃₃D₄ C₄²⋊₁₀D₄ C₄².₁₃₀D₄ ...
(C₂×C₈).D_2p: M₄(2).₄₆D₄ C₄.(C₄×D₄) M₄(2).D₄ (C₂×C₈).₆D₄ C₈.D₄⋊C₂ C₂³.₅₁D₁₂ C₂³.₄₆D₂₀ C₂³.₄₆D₂₈ ...
C_4p.(C₂×D₄): M₄(2)⋊₁₅D₄ (C₂×C₈)⋊₁₁D₄ (C₆×Q₈)⋊₆C₄ (Q₈×C₁₀)⋊₁₆C₄ (Q₈×C₁₄)⋊₆C₄ ...
C₂³.₃₈D₄ is a maximal quotient of
C₄².₃₉₉D₄ C₄².₄₀₁D₄ Q₈⋊₅M₄(2) C₄².₅₄D₄ C₄².₅₆D₄ C₂⁴.₅₅D₄ C₂⁴.₅₇D₄ C₄².₆₀D₄ C₄².₄₁₅D₄ C₄².₄₁₈D₄ C₄².₈₃D₄ C₄².₈₅D₄ C₄².₈₆D₄ C₂⁴.₁₅₂D₄ Q₈⋊C₄² C₂⁴.₁₅₅D₄ C₄².₁₀₁D₄ C₄².₁₂₂D₄ C₄².₁₂₅D₄ (C₂×Q₈)⋊₄F₅
C₂³.D_4p: C₂³.₃₆D₈ C₂³.₅₁D₁₂ C₂³.₄₆D₂₀ C₂³.₄₆D₂₈ ...
C₄.(C₂×D_4p): C₄².₄₁₄D₄ C₄⋊C₄.₂₃₇D₆ C₄.(C₂×D₂₀) (C₂×C₄).₄₇D₂₈ ...
(C₂×C_4p).D₄: C₂⁴.₇₅D₄ C₄².₁₁₁D₄ (C₆×Q₈)⋊₆C₄ (Q₈×C₁₀)⋊₁₆C₄ (Q₈×C₁₄)⋊₆C₄ ...
C₄⋊C₄.D_2p: C₂⁴.₇₃D₄ C₄².₁₁₇D₄ (S₃×Q₈)⋊C₄ (Q₈×D₅)⋊C₄ (Q₈×D₇)⋊C₄ ...

Matrix representation of C₂³.₃₈D₄ ►in GL₆(𝔽₁₇)

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

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

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

0	4	0	0	0	0
13	0	0	0	0	0
0	0	0	0	0	4
0	0	0	0	4	0
0	0	4	0	0	0
0	0	0	13	0	0

0	4	0	0	0	0
4	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,4,0,0,0,0,4,0,0,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂³.₃₈D₄ in GAP, Magma, Sage, TeX

C_2^3._{38}D_4

% in TeX

G:=Group("C2^3.38D4");

// GroupNames label

G:=SmallGroup(64,100);

// by ID

G=gap.SmallGroup(64,100);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,332,158,963,489,117]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.₃₈D₄ in TeX

G = C23.38D4 order 64 = 26

9th non-split extension by C23 of D4 acting via D4/C22=C2

G = C₂³.₃₈D₄ order 64 = 2⁶

9^th non-split extension by C₂³ of D₄ acting via D₄/C₂²=C₂