C4^2.352C2^3 - GroupNames

G = C₄².₃₅₂C₂³ order 128 = 2⁷

213^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₅₂C₂³, (C₄×D₈)⋊₂₂C₂, C₄⋊Q₈⋊₈C₂², D₈⋊C₄⋊₉C₂, C₄⋊C₈⋊₁₄C₂², (C₄×C₈)⋊₁₅C₂², C₄⋊C₄.₃₄₅D₄, D₄⋊₂Q₈⋊₇C₂, (C₄×SD₁₆)⋊₅C₂, (C₄×D₄)⋊₉C₂², (C₄×Q₈)⋊₉C₂², C₈⋊C₄⋊₅C₂², D₄⋊Q₈⋊₂₄C₂, D₄.₉(C₄○D₄), C₂²⋊SD₁₆⋊₆C₂, C₂²⋊D₈.₄C₂, C₂.₁₇(D₄○D₈), C₄⋊C₄.₇₁C₂³, (C₂×C₈).₄₅C₂³, C₄.Q₈⋊₆₇C₂², C₂.D₈⋊₂₅C₂², D₄.₂D₄⋊₂₀C₂, (C₂×C₄).₃₁₆C₂⁴, C₂²⋊C₄.₁₄₆D₄, C₄.₄D₄⋊₇C₂², C₂³.₂₅₅(C₂×D₄), SD₁₆⋊C₄⋊₁₂C₂, (C₂×Q₈).₈₀C₂³, D₄⋊C₄⋊₈₀C₂², C₂.₂₆(D₄○SD₁₆), Q₈⋊C₄⋊₂₅C₂², (C₂×D₄).₄₀₇C₂³, (C₂×D₈).₁₂₅C₂², C₄⋊D₄.₂₆C₂², C₂²⋊C₈.₂₉C₂², C₂².₁₁C₂⁴⋊₁₁C₂, C₂²⋊Q₈.₂₆C₂², C₂³.₂₀D₄⋊₁₉C₂, C₂³.₁₉D₄⋊₁₉C₂, (C₂²×C₄).₂₈₉C₂³, C₄².₇C₂²⋊₁C₂, (C₂×SD₁₆).₁₇C₂², C₂².₅₇₆(C₂²×D₄), C₂².₃₆C₂⁴⋊₁C₂, (C₂²×D₄).₃₆₁C₂², C₄²⋊C₂.₁₂₇C₂², C₂.₁₁₇(C₂².₁₉C₂⁴), C₄.₂₀₁(C₂×C₄○D₄), (C₂×C₄).₅₀₀(C₂×D₄), SmallGroup(128,1850)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₅₂C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×D₄ — C₂².₁₁C₂⁴ — C₄².₃₅₂C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₅₂C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₅₂C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₅₂C₂³

Generators and relations for C₄².₃₅₂C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=d²=e²=1, ab=ba, ac=ca, dad=ab², ae=ea, cbc=ebe=b^-1, bd=db, dcd=a²c, ece=bc, de=ed >

Subgroups: 436 in 203 conjugacy classes, 88 normal (84 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, 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, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄⋊Q₈, C₂×D₈, C₂×SD₁₆, C₂²×D₄, C₄².₇C₂², C₄×D₈, C₄×SD₁₆, SD₁₆⋊C₄, D₈⋊C₄, C₂²⋊D₈, C₂²⋊SD₁₆, D₄.₂D₄, D₄⋊Q₈, D₄⋊₂Q₈, C₂³.₁₉D₄, C₂³.₂₀D₄, C₂².₁₁C₂⁴, C₂².₃₆C₂⁴, C₄².₃₅₂C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○D₈, D₄○SD₁₆, C₄².₃₅₂C₂³

Smallest permutation representation of C₄².₃₅₂C₂³
►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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 32 9)(6 16 29 10)(7 13 30 11)(8 14 31 12)

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

(2 26)(4 28)(5 30)(6 8)(7 32)(9 13)(10 12)(11 15)(14 16)(18 24)(20 22)(29 31)

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○D₈

D₄○SD₁₆

kernel

C₄².₃₅₂C₂³

C₄².₇C₂²

C₄×D₈

C₄×SD₁₆

SD₁₆⋊C₄

D₈⋊C₄

C₂²⋊D₈

C₂²⋊SD₁₆

D₄.₂D₄

D₄⋊Q₈

D₄⋊₂Q₈

C₂³.₁₉D₄

C₂³.₂₀D₄

C₂².₁₁C₂⁴

C₂².₃₆C₂⁴

C₂²⋊C₄

C₄⋊C₄

D₄

C₂

# reps

Matrix representation of C₄².₃₅₂C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	0	1	0
0	0	16	1	1	15
0	0	1	0	0	0
0	0	0	0	0	16

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	16	1	1	15
0	0	16	0	1	16

6	15	0	0	0	0
9	11	0	0	0	0
0	0	3	14	0	0
0	0	14	14	0	0
0	0	3	14	0	6
0	0	14	0	3	0

16	0	0	0	0	0
11	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	16	1	0	16

1	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	1	0
0	0	16	0	1	16

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,15,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,16,0,0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,15,16],[6,9,0,0,0,0,15,11,0,0,0,0,0,0,3,14,3,14,0,0,14,14,14,0,0,0,0,0,0,3,0,0,0,0,6,0],[16,11,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,1,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;

C₄².₃₅₂C₂³ in GAP, Magma, Sage, TeX

C_4^2._{352}C_2^3

% in TeX

G:=Group("C4^2.352C2^3");

// GroupNames label

G:=SmallGroup(128,1850);

// by ID

G=gap.SmallGroup(128,1850);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,1018,80,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.352C23 order 128 = 27

213rd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₅₂C₂³ order 128 = 2⁷

213^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²