C4^2.18C2^3 - GroupNames

G = C₄².₁₈C₂³ order 128 = 2⁷

18^th non-split extension by C₄² of C₂³ acting faithfully

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

Aliases: C₄².₁₈C₂³, C₄oD₄:₇D₄, C₄:C₈:₈C₂², D₄.₉(C₂xD₄), Q₈.₉(C₂xD₄), C₄:D₈:₂₀C₂, C₄:C₄.₃₃₉D₄, C₄:SD₁₆:₄C₂, C₂.₉(D₄oD₈), (C₄xD₄):₄C₂², (C₄xQ₈):₄C₂², (C₂xD₈):₄₁C₂², (C₂xC₈).₂₀C₂³, C₄.₇₆(C₂²xD₄), D₄.₂D₄:₁₆C₂, C₄:C₄.₃₈₆C₂³, (C₂xC₄).₂₄₉C₂⁴, Q₈.D₄:₁₆C₂, C₂²:C₄.₁₄₀D₄, (C₂xQ₁₆):₄₁C₂², (C₂xD₄).₅₅C₂³, C₂³.₄₄₆(C₂xD₄), (C₂xQ₈).₄₂C₂³, C₄.₁₇₀(C₄:D₄), D₄:C₄:₁₇C₂², C₂².₂₉C₂⁴:₉C₂, C₂.₁₄(D₄oSD₁₆), Q₈:C₄:₁₉C₂², (C₂xSD₁₆):₇₄C₂², C₄:₁D₄.₅₉C₂², C₂³.₃₆D₄:₇C₂, C₂².₆(C₄:D₄), (C₂²xC₄).₉₇₉C₂³, (C₂²xC₈).₁₇₈C₂², C₄².₆C₂²:₇C₂, C₄.₄D₄.₂₆C₂², C₂².₅₀₉(C₂²xD₄), C₂³.₃₃C₂³:₇C₂, (C₂²xD₄).₃₄₃C₂², (C₂xM₄(2)).₅₆C₂², C₄²:C₂.₁₀₄C₂², (C₂xC₄oD₈):₆C₂, (C₂xC₈:C₂²):₁₇C₂, C₄.₁₅₉(C₂xC₄oD₄), (C₂xC₄).₄₆₉(C₂xD₄), C₂.₆₇(C₂xC₄:D₄), (C₂xD₄:C₄):₂₉C₂, (C₂xC₄).₂₈₀(C₄oD₄), (C₂xC₄:C₄).₅₈₃C₂², (C₂xC₄oD₄).₁₂₁C₂², SmallGroup(128,1777)

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

Derived series

C₁ — C₂xC₄ — C₄².₁₈C₂³

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂xC₄oD₄ — C₂³.₃₃C₂³ — C₄².₁₈C₂³

Lower central

C₁ — C₂ — C₂xC₄ — C₄².₁₈C₂³

Upper central

C₁ — C₂² — C₄²:C₂ — C₄².₁₈C₂³

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₄².₁₈C₂³

Generators and relations for C₄².₁₈C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=d²=1, e²=a², ab=ba, cac=a^-1, ad=da, eae^-1=ab², cbc=dbd=b^-1, be=eb, dcd=bc, ece^-1=a²c, de=ed >

Subgroups: 548 in 252 conjugacy classes, 100 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₄:C₄, C₄:C₄, C₂xC₈, C₂xC₈, C₂xC₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, D₄:C₄, Q₈:C₄, C₄:C₈, C₂xC₄:C₄, C₂xC₄:C₄, C₄²:C₂, C₄²:C₂, C₄xD₄, C₄xD₄, C₄xQ₈, C₂²wrC₂, C₄:D₄, C₄.₄D₄, C₄:₁D₄, C₂²xC₈, C₂xM₄(2), C₂xD₈, C₂xD₈, C₂xSD₁₆, C₂xSD₁₆, C₂xQ₁₆, C₄oD₈, C₈:C₂², C₂²xD₄, C₂xC₄oD₄, C₂xD₄:C₄, C₂³.₃₆D₄, C₄².₆C₂², C₄:D₈, C₄:SD₁₆, D₄.₂D₄, Q₈.D₄, C₂³.₃₃C₂³, C₂².₂₉C₂⁴, C₂xC₄oD₈, C₂xC₈:C₂², C₄².₁₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₄oD₄, C₂⁴, C₄:D₄, C₂²xD₄, C₂xC₄oD₄, C₂xC₄:D₄, D₄oD₈, D₄oSD₁₆, C₄².₁₈C₂³

Smallest permutation representation of C₄².₁₈C₂³
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	4A	4B	4C	4D	4E	···	4N	4O	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	8	8	8	8	8	8
size	1	1	1	1	2	2	4	4	8	8	8	2	2	2	2	4	···	4	8	4	4	4	4	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄oD₄

D₄oD₈

D₄oSD₁₆

kernel

C₄².₁₈C₂³

C₂xD₄:C₄

C₂³.₃₆D₄

C₄².₆C₂²

C₄:D₈

C₄:SD₁₆

D₄.₂D₄

Q₈.D₄

C₂³.₃₃C₂³

C₂².₂₉C₂⁴

C₂xC₄oD₈

C₂xC₈:C₂²

C₂²:C₄

C₄:C₄

C₄oD₄

C₂xC₄

C₂

# reps

Matrix representation of C₄².₁₈C₂³ ►in GL₆(F₁₇)

13	8	0	0	0	0
0	4	0	0	0	0
0	0	10	0	10	10
0	0	5	0	0	10
0	0	0	12	12	12
0	0	12	5	12	12

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

4	9	0	0	0	0
4	13	0	0	0	0
0	0	0	0	7	10
0	0	5	0	0	10
0	0	5	12	5	12
0	0	0	12	5	12

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

4	9	0	0	0	0
0	13	0	0	0	0
0	0	0	0	7	10
0	0	12	0	7	0
0	0	0	12	5	12
0	0	5	12	5	12

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,8,4,0,0,0,0,0,0,10,5,0,12,0,0,0,0,12,5,0,0,10,0,12,12,0,0,10,10,12,12],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0,0,0,16,0],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,5,5,0,0,0,0,0,12,12,0,0,7,0,5,5,0,0,10,10,12,12],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,12,0,5,0,0,0,0,12,12,0,0,7,7,5,5,0,0,10,0,12,12] >;

C₄².₁₈C₂³ in GAP, Magma, Sage, TeX

C_4^2._{18}C_2^3

% in TeX

G:=Group("C4^2.18C2^3");

// GroupNames label

G:=SmallGroup(128,1777);

// by ID

G=gap.SmallGroup(128,1777);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,248,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.18C23 order 128 = 27

18th non-split extension by C42 of C23 acting faithfully

G = C₄².₁₈C₂³ order 128 = 2⁷

18^th non-split extension by C₄² of C₂³ acting faithfully