C2^3.46D4 - GroupNames

G = C₂³.₄₆D₄ order 64 = 2⁶

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

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

Aliases: C₂³.₄₆D₄, C₂².₈SD₁₆, C₄.Q₈⋊₉C₂, (C₂×C₄).₃₈D₄, C₂²⋊C₈⋊₁₀C₂, C₄⋊D₄.₆C₂, D₄⋊C₄⋊₁₂C₂, C₄.₃₀(C₄○D₄), C₄⋊C₄.₆₄C₂², (C₂×C₈).₃₇C₂², C₂.₁₂(C₂×SD₁₆), C₂.₁₇(C₈⋊C₂²), (C₂×C₄).₁₀₆C₂³, (C₂×D₄).₂₂C₂², C₂².₁₀₂(C₂×D₄), (C₂²×C₄).₅₂C₂², C₂.₁₂(C₂².D₄), (C₂×C₄⋊C₄)⋊₁₂C₂, SmallGroup(64,162)

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

Derived series

C₁ — C₂×C₄ — C₂³.₄₆D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄⋊C₄ — C₂×C₄⋊C₄ — C₂³.₄₆D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂³.₄₆D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂³.₄₆D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂³.₄₆D₄

Generators and relations for C₂³.₄₆D₄
G = < a,b,c,d,e | a²=b²=c²=e²=1, d⁴=c, dad^-1=eae=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd³ >

Subgroups: 113 in 57 conjugacy classes, 27 normal (15 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂²⋊C₈, D₄⋊C₄, C₄.Q₈, C₂×C₄⋊C₄, C₄⋊D₄, C₂³.₄₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₂².D₄, C₂×SD₁₆, C₈⋊C₂², C₂³.₄₆D₄

Character table of C₂³.₄₆D₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D
size	1	1	1	1	2	2	8	2	2	4	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	-2	0	-2	-2	0	0	0	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	0	-2	-2	0	0	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	0	0	0	-2	2	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₂	2	-2	2	-2	0	0	0	2	-2	0	0	-2i	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₃	2	-2	2	-2	0	0	0	2	-2	0	0	2i	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₄	2	-2	2	-2	0	0	0	-2	2	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	√-2	-√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₆	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₇	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₈	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	-√-2	√-2	-√-2	√-2	complex lifted from SD₁₆
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 C₂³.₄₆D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

C₂³.₄₆D₄ is a maximal subgroup of
C₂⁴.₁₁₅D₄ C₂⁴.₁₈₃D₄ C₂⁴.₁₁₇D₄ (C₂×D₄).₃₀₁D₄ (C₂×D₄).₃₀₄D₄ C₄².₂₂₅D₄ C₄².₂₂₇D₄ C₄².₂₃₅D₄ C₂³⋊₄SD₁₆ C₂⁴.₁₂₁D₄ C₂⁴.₁₂₅D₄ C₂⁴.₁₂₉D₄ C₄.2₊¹⁺⁴ C₄.₁₉2₊¹⁺⁴ C₄².₂₈₄D₄ C₄².₂₈₆D₄ C₄².₂₉₁D₄
C₄⋊C₄.D_2p: C₂⁴.₁₆D₄ C₄⋊C₄.₁₈D₄ C₄⋊C₄.₁₉D₄ C₂⁴.₁₈D₄ C₄².₃₅₃C₂³ C₄².₃₅₉C₂³ C₄².₄₂₃C₂³ C₄².₄₂₆C₂³ ...
C_2p.(C₂×SD₁₆): C₄².₂₂₃D₄ C₄².₂₇₉D₄ C₂³.₄₃D₁₂ C₂³.₃₈D₂₀ C₂³.₃₈D₂₈ ...
C₂³.₄₆D₄ is a maximal quotient of
C₂⁴.₁₅₇D₄ C₄.Q₈⋊₉C₄ (C₂×C₄).₂₄D₈ C₂⁴.₈₉D₄ (C₂×C₈).₁₆₉D₄ (C₂×C₄).₂₃Q₁₆ M₅(2).C₂² C₂³.₁₀SD₁₆
C₂³.D_4p: C₂³.₃₈D₈ C₂³.₄₃D₁₂ C₂³.₃₈D₂₀ C₂³.₃₈D₂₈ ...
C₄⋊C₄.D_2p: C₂⁴.₁₅₉D₄ C₄.₆₇(C₄×D₄) C₂⁴.₈₄D₄ C₂.(C₈⋊₃Q₈) D₆.SD₁₆ D₆.₄SD₁₆ C₄⋊C₄.₂₂₈D₆ C₄⋊D₄.S₃ ...

Matrix representation of C₂³.₄₆D₄ ►in GL₄(𝔽₁₇) generated by

0	1	0	0
1	0	0	0
0	0	1	0
0	0	0	1

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

4	0	0	0
0	13	0	0
0	0	12	5
0	0	12	12

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

G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[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],[4,0,0,0,0,13,0,0,0,0,12,12,0,0,5,12],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C₂³.₄₆D₄ in GAP, Magma, Sage, TeX

C_2^3._{46}D_4

% in TeX

G:=Group("C2^3.46D4");

// GroupNames label

G:=SmallGroup(64,162);

// by ID

G=gap.SmallGroup(64,162);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,194,1444,376,88]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.₄₆D₄ in TeX

G = C23.46D4 order 64 = 26

17th non-split extension by C23 of D4 acting via D4/C22=C2

G = C₂³.₄₆D₄ order 64 = 2⁶

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