C4^2.667C2^3 - GroupNames

G = C₄².₆₆₇C₂³ order 128 = 2⁷

82^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂²=C₂

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

Aliases: C₄².₆₆₇C₂³, C₈⋊C₈⋊₁₃C₂, (C₂×C₈).₃₆D₄, C₈⋊₃Q₈⋊₁₉C₂, C₄⋊Q₁₆⋊₅C₂, C₄.₁₀(C₄○D₈), C₄.₅(C₈⋊C₂²), C₄.₄D₈.₆C₂, C₄⋊Q₈.₉₁C₂², C₂.₁₀(C₈⋊₃D₄), (C₄×C₈).₂₆₀C₂², C₄.SD₁₆⋊₃₉C₂, C₄.₅(C₈.C₂²), C₂.₁₀(C₈.₂D₄), C₄⋊₁D₄.₅₀C₂², C₂.₁₅(C₈.₁₂D₄), C₂².₆₈(C₄⋊₁D₄), (C₂×C₄).₇₂₄(C₂×D₄), SmallGroup(128,452)

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

Derived series

C₁ — C₄² — C₄².₆₆₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×C₈ — C₈⋊C₈ — C₄².₆₆₇C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₆₆₇C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₆₆₇C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₆₆₇C₂³

Generators and relations for C₄².₆₆₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=a², d²=a^-1b², e²=b, ab=ba, cac^-1=a^-1, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, dcd^-1=a^-1c, ece^-1=b^-1c, ede^-1=a²d >

Subgroups: 240 in 93 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄⋊C₄, C₂×C₈, Q₁₆, C₂×D₄, C₂×Q₈, C₄×C₈, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₄⋊₁D₄, C₄⋊Q₈, C₂×Q₁₆, C₈⋊C₈, C₄.₄D₈, C₄.SD₁₆, C₄⋊Q₁₆, C₈⋊₃Q₈, C₄².₆₆₇C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄⋊₁D₄, C₄○D₈, C₈⋊C₂², C₈.C₂², C₈.₁₂D₄, C₈⋊₃D₄, C₈.₂D₄, C₄².₆₆₇C₂³

Character table of C₄².₆₆₇C₂³

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L
size	1	1	1	1	16	2	2	2	2	2	2	16	16	16	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	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	0	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	2	0	0	-2	0	0	-2	2	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	2	-2	2	-2	-2	-2	0	0	0	2	-2	0	0	0	-2	0	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	-2	-2	-2	2	-2	2	0	0	0	0	0	-2	-2	0	0	2	0	0	2	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	-2	-2	-2	2	-2	2	0	0	0	0	0	2	2	0	0	-2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	-2	0	0	2	0	0	2	-2	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	0	2	-2	2	-2	-2	-2	0	0	0	-2	2	0	0	0	2	0	0	-2	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	2	-2	0	0	-2	0	0	2	0	0	0	0	√2	√2	-√-2	√-2	0	-√2	√-2	-2i	-√2	-√-2	2i	0	complex lifted from C₄○D₈
ρ₁₆	2	-2	2	-2	0	0	-2	0	0	2	0	0	0	0	-√2	-√2	√-2	-√-2	0	√2	-√-2	-2i	√2	√-2	2i	0	complex lifted from C₄○D₈
ρ₁₇	2	-2	2	-2	0	0	-2	0	0	2	0	0	0	0	-√2	-√2	-√-2	√-2	0	√2	√-2	2i	√2	-√-2	-2i	0	complex lifted from C₄○D₈
ρ₁₈	2	-2	2	-2	0	0	2	0	0	-2	0	0	0	0	√2	-√2	√-2	-√-2	-2i	√2	√-2	0	-√2	-√-2	0	2i	complex lifted from C₄○D₈
ρ₁₉	2	-2	2	-2	0	0	2	0	0	-2	0	0	0	0	-√2	√2	√-2	-√-2	2i	-√2	√-2	0	√2	-√-2	0	-2i	complex lifted from C₄○D₈
ρ₂₀	2	-2	2	-2	0	0	2	0	0	-2	0	0	0	0	-√2	√2	-√-2	√-2	-2i	-√2	-√-2	0	√2	√-2	0	2i	complex lifted from C₄○D₈
ρ₂₁	2	-2	2	-2	0	0	-2	0	0	2	0	0	0	0	√2	√2	√-2	-√-2	0	-√2	-√-2	2i	-√2	√-2	-2i	0	complex lifted from C₄○D₈
ρ₂₂	2	-2	2	-2	0	0	2	0	0	-2	0	0	0	0	√2	-√2	-√-2	√-2	2i	√2	-√-2	0	-√2	√-2	0	-2i	complex lifted from C₄○D₈
ρ₂₃	4	-4	-4	4	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₄	4	-4	-4	4	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₅	4	4	-4	-4	0	0	0	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₆	4	4	-4	-4	0	0	0	0	-4	0	4	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 C₄².₆₆₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

Matrix representation of C₄².₆₆₇C₂³ ►in GL₆(𝔽₁₇)

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

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

0	6	0	0	0	0
3	0	0	0	0	0
0	0	9	4	0	16
0	0	3	8	1	1
0	0	8	8	5	0
0	0	9	0	4	12

4	8	0	0	0	0
13	13	0	0	0	0
0	0	9	9	12	0
0	0	3	4	5	10
0	0	8	13	0	1
0	0	9	13	8	4

10	10	0	0	0	0
12	0	0	0	0	0
0	0	0	0	1	0
0	0	1	1	16	15
0	0	0	1	0	0
0	0	1	0	0	16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1,16,0,0,1,0,1,0,0,0,0,0,16,1,0,0,0,0,15,1],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,1,16,0,0,1,0,1,0,0,0,0,0,16,1,0,0,0,0,15,1],[0,3,0,0,0,0,6,0,0,0,0,0,0,0,9,3,8,9,0,0,4,8,8,0,0,0,0,1,5,4,0,0,16,1,0,12],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,9,3,8,9,0,0,9,4,13,13,0,0,12,5,0,8,0,0,0,10,1,4],[10,12,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,1,0,0,0,1,16,0,0,0,0,0,15,0,16] >;

C₄².₆₆₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{667}C_2^3

% in TeX

G:=Group("C4^2.667C2^3");

// GroupNames label

G:=SmallGroup(128,452);

// by ID

G=gap.SmallGroup(128,452);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,736,422,387,100,1123,136,2804,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄².₆₆₇C₂³ in TeX

G = C42.667C23 order 128 = 27

82nd non-split extension by C42 of C23 acting via C23/C22=C2

G = C₄².₆₆₇C₂³ order 128 = 2⁷

82^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂²=C₂