C4^2.356C2^3 - GroupNames

G = C₄².₃₅₆C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₅₆C₂³, (C₄×D₈)⋊₅C₂, (C₄×C₈)⋊₈C₂², C₄⋊D₈⋊₂₄C₂, C₄⋊C₈⋊₁₅C₂², C₄⋊C₄.₃₄₉D₄, D₄.Q₈⋊₂₁C₂, C₂²⋊D₈⋊₁₆C₂, D₈⋊C₄⋊₁₁C₂, C₈⋊C₄⋊₆C₂², C₂.₁₉(D₄○D₈), (C₄×D₄)⋊₁₀C₂², C₄⋊₁D₄⋊₆C₂², C₄⋊C₄.₇₅C₂³, (C₂×C₈).₄₉C₂³, C₂.D₈⋊₅₈C₂², C₄.Q₈⋊₁₄C₂², D₄.₁₂(C₄○D₄), (C₂×C₄).₃₂₀C₂⁴, C₂²⋊C₄.₁₅₀D₄, C₂³.₂₅₉(C₂×D₄), C₄².C₂⋊₂C₂², D₄⋊C₄⋊₂₃C₂², (C₂×D₈).₁₂₇C₂², (C₂×D₄).₄₁₀C₂³, C₄⋊D₄.₂₉C₂², C₂²⋊C₈.₃₃C₂², C₂².₁₁C₂⁴⋊₁₂C₂, C₂³.₁₉D₄⋊₂₀C₂, (C₂²×C₄).₂₉₃C₂³, C₄².₇C₂²⋊₅C₂, C₂².₅₈₀(C₂²×D₄), C₂².₃₄C₂⁴⋊₂C₂, (C₂²×D₄).₃₆₂C₂², C₄²⋊C₂.₁₃₁C₂², C₂.₁₂₁(C₂².₁₉C₂⁴), C₄.₂₀₅(C₂×C₄○D₄), (C₂×C₄).₅₀₄(C₂×D₄), SmallGroup(128,1854)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₅₆C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×D₄ — C₂².₁₁C₂⁴ — C₄².₃₅₆C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₅₆C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₅₆C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₅₆C₂³

Generators and relations for C₄².₃₅₆C₂³
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=a²b²c, ece=bc, de=ed >

Subgroups: 476 in 209 conjugacy classes, 88 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, D₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, D₄⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄⋊D₄, C₄⋊D₄, C₂².D₄, C₄².C₂, C₄⋊₁D₄, C₂×D₈, C₂²×D₄, C₄².₇C₂², C₄×D₈, D₈⋊C₄, C₂²⋊D₈, C₄⋊D₈, D₄.Q₈, C₂³.₁₉D₄, C₂².₁₁C₂⁴, C₂².₃₄C₂⁴, C₄².₃₅₆C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○D₈, C₄².₃₅₆C₂³

Smallest permutation representation of C₄².₃₅₆C₂³
►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 32)(2 29)(3 30)(4 31)(5 26)(6 27)(7 28)(8 25)(9 19)(10 20)(11 17)(12 18)(13 23)(14 24)(15 21)(16 22)

(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), (1,32)(2,29)(3,30)(4,31)(5,26)(6,27)(7,28)(8,25)(9,19)(10,20)(11,17)(12,18)(13,23)(14,24)(15,21)(16,22), (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), (1,32)(2,29)(3,30)(4,31)(5,26)(6,27)(7,28)(8,25)(9,19)(10,20)(11,17)(12,18)(13,23)(14,24)(15,21)(16,22), (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)], [(1,32),(2,29),(3,30),(4,31),(5,26),(6,27),(7,28),(8,25),(9,19),(10,20),(11,17),(12,18),(13,23),(14,24),(15,21),(16,22)], [(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	2I	2J	4A	···	4F	4G	···	4M	4N	4O	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	···	2	2	2	4	···	4	4	···	4	4	4	8	8	8	8	8	8
size	1	1	1	1	4	···	4	8	8	2	···	2	4	···	4	8	8	4	4	4	4	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○D₈

kernel

C₄².₃₅₆C₂³

C₄².₇C₂²

C₄×D₈

D₈⋊C₄

C₂²⋊D₈

C₄⋊D₈

D₄.Q₈

C₂³.₁₉D₄

C₂².₁₁C₂⁴

C₂².₃₄C₂⁴

C₂²⋊C₄

C₄⋊C₄

D₄

C₂

# reps

Matrix representation of C₄².₃₅₆C₂³ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	13	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

16	0	0	0	0	0
0	16	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

0	15	0	0	0	0
8	0	0	0	0	0
0	0	12	5	11	6
0	0	5	5	6	6
0	0	14	3	5	12
0	0	3	3	12	12

16	0	0	0	0	0
0	1	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	13	0	1	0
0	0	0	13	0	1

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

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,16,0,0,0,0,4,0,16,0,0,15,0,13,0,0,0,0,15,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,12,5,14,3,0,0,5,5,3,3,0,0,11,6,5,12,0,0,6,6,12,12],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,13,0,0,0,0,16,0,13,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₄².₃₅₆C₂³ in GAP, Magma, Sage, TeX

C_4^2._{356}C_2^3

% in TeX

G:=Group("C4^2.356C2^3");

// GroupNames label

G:=SmallGroup(128,1854);

// by ID

G=gap.SmallGroup(128,1854);

# by ID

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

// 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*b^2*c,e*c*e=b*c,d*e=e*d>;

// generators/relations

G = C42.356C23 order 128 = 27

217th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₅₆C₂³ order 128 = 2⁷

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