C4^2.357C2^3 - GroupNames

G = C₄².₃₅₇C₂³ order 128 = 2⁷

218^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₅₇C₂³, C₄⋊Q₈⋊₉C₂², C₄⋊C₈⋊₁₆C₂², (C₄×C₈)⋊₄₁C₂², C₄⋊C₄.₃₅₀D₄, D₄.Q₈⋊₂₂C₂, C₈⋊C₄⋊₇C₂², (C₄×SD₁₆)⋊₃₀C₂, D₄.D₄⋊₈C₂, C₄⋊C₄.₇₆C₂³, (C₂×C₈).₅₀C₂³, (C₄×Q₈)⋊₁₀C₂², C₂.D₈⋊₂₆C₂², C₄.Q₈⋊₆₈C₂², D₄.₁₃(C₄○D₄), (C₂×C₄).₃₂₁C₂⁴, C₂²⋊C₄.₁₅₁D₄, C₂²⋊SD₁₆.₂C₂, (C₄×D₄).₈₃C₂², C₂³.₂₆₀(C₂×D₄), SD₁₆⋊C₄⋊₁₆C₂, (C₂×Q₈).₈₂C₂³, C₄².C₂⋊₃C₂², C₂.₃₀(D₄○SD₁₆), Q₈⋊C₄⋊₂₆C₂², (C₂×D₄).₄₁₁C₂³, C₂²⋊C₈.₃₄C₂², C₂²⋊Q₈.₂₉C₂², D₄⋊C₄.₃₇C₂², C₂³.₂₀D₄⋊₂₁C₂, (C₂²×C₄).₂₉₄C₂³, C₄².₇C₂²⋊₆C₂, C₂².₁₁C₂⁴.₉C₂, C₂².₅₈₁(C₂²×D₄), C₂².₃₅C₂⁴⋊₂C₂, (C₂×SD₁₆).₁₄₅C₂², (C₂²×D₄).₃₆₃C₂², C₄²⋊C₂.₁₃₂C₂², C₂.₁₂₂(C₂².₁₉C₂⁴), C₄.₂₀₆(C₂×C₄○D₄), (C₂×C₄).₅₀₅(C₂×D₄), SmallGroup(128,1855)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₅₇C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×D₄ — C₂².₁₁C₂⁴ — C₄².₃₅₇C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₅₇C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₅₇C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₅₇C₂³

Generators and relations for C₄².₃₅₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=e²=1, c²=b², ab=ba, ac=ca, dad=ab², ae=ea, cbc^-1=ebe=b^-1, bd=db, dcd=a²b²c, ece=bc, de=ed >

Subgroups: 396 in 197 conjugacy classes, 88 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂×SD₁₆, C₂²×D₄, C₄².₇C₂², C₄×SD₁₆, SD₁₆⋊C₄, C₂²⋊SD₁₆, D₄.D₄, D₄.Q₈, C₂³.₂₀D₄, C₂².₁₁C₂⁴, C₂².₃₅C₂⁴, C₄².₃₅₇C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○SD₁₆, C₄².₃₅₇C₂³

Smallest permutation representation of C₄².₃₅₇C₂³
►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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 30 15 12)(6 31 16 9)(7 32 13 10)(8 29 14 11)

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

(2 17)(4 19)(5 13)(6 8)(7 15)(9 11)(10 30)(12 32)(14 16)(22 27)(24 25)(29 31)

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2H	4A	···	4F	4G	···	4M	4N	4O	4P	4Q	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	···	2	4	···	4	4	···	4	4	4	4	4	8	8	8	8	8	8
size	1	1	1	1	4	···	4	2	···	2	4	···	4	8	8	8	8	4	4	4	4	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

kernel

C₄².₃₅₇C₂³

C₄².₇C₂²

C₄×SD₁₆

SD₁₆⋊C₄

C₂²⋊SD₁₆

D₄.D₄

D₄.Q₈

C₂³.₂₀D₄

C₂².₁₁C₂⁴

C₂².₃₅C₂⁴

C₂²⋊C₄

C₄⋊C₄

D₄

C₂

# reps

Matrix representation of C₄².₃₅₇C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	15	0
0	0	0	4	0	15
0	0	16	0	13	0
0	0	0	16	0	13

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

1	2	0	0	0	0
0	16	0	0	0	0
0	0	14	3	10	7
0	0	3	3	7	7
0	0	5	12	3	14
0	0	12	12	14	14

1	0	0	0	0	0
16	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	4	0	16	0
0	0	0	4	0	16

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

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,16,0,0,0,0,4,0,16,0,0,15,0,13,0,0,0,0,15,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,14,3,5,12,0,0,3,3,12,12,0,0,10,7,3,14,0,0,7,7,14,14],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,4,0,0,0,0,1,0,4,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

C₄².₃₅₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{357}C_2^3

% in TeX

G:=Group("C4^2.357C2^3");

// GroupNames label

G:=SmallGroup(128,1855);

// by ID

G=gap.SmallGroup(128,1855);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,1018,304,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.357C23 order 128 = 27

218th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₅₇C₂³ order 128 = 2⁷

218^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²