C2^3:C8:C2 - GroupNames

G = C₂³⋊C₈⋊C₂ order 128 = 2⁷

15^th semidirect product of C₂³⋊C₈ and C₂ acting faithfully

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

Aliases: C₂³⋊C₈⋊₁₅C₂, C₂⁴.₁(C₂×C₄), (C₂²×D₄).₄C₄, C₂².₉(C₈○D₄), (C₂²×C₄).₆₅₁D₄, C₂⁴.₄C₄⋊₁₇C₂, C₂².₅(C₂³⋊C₄), C₂²⋊C₈.₁₂₄C₂², C₂³.₉₁(C₂²⋊C₄), (C₂³×C₄).₂₀₀C₂², C₂³.₁₆₉(C₂²×C₄), (C₂²×C₄).₄₃₂C₂³, C₂².M₄(2)⋊₁₅C₂, C₂.₈(M₄(2).₈C₂²), (C₂×C₄⋊C₄).₁₁C₄, (C₂×C₂²⋊C₈)⋊₃C₂, C₂.₉(C₂×C₂³⋊C₄), (C₂×C₄⋊C₄).₉C₂², (C₂×C₂²⋊C₄).₇C₄, (C₂²×C₄).₉(C₂×C₄), (C₂×C₄).₁₁₂₉(C₂×D₄), (C₂×C₄).₇₀(C₂²⋊C₄), C₂.₆((C₂²×C₈)⋊C₂), (C₂×C₂²⋊C₄).₈₆C₂², C₂².₁₅₀(C₂×C₂²⋊C₄), (C₂×C₂².D₄).₁C₂, SmallGroup(128,200)

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

Derived series

C₁ — C₂³ — C₂³⋊C₈⋊C₂

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₂².D₄ — C₂³⋊C₈⋊C₂

Lower central

C₁ — C₂ — C₂³ — C₂³⋊C₈⋊C₂

Upper central

C₁ — C₂² — C₂³×C₄ — C₂³⋊C₈⋊C₂

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂³⋊C₈⋊C₂

Generators and relations for C₂³⋊C₈⋊C₂
G = < a,b,c,d,e | a²=b²=c²=d⁸=e²=1, ab=ba, ac=ca, dad^-1=abc, eae=ad⁴, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 324 in 140 conjugacy classes, 46 normal (34 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₂²×C₈, C₂×M₄(2), C₂³×C₄, C₂²×D₄, C₂³⋊C₈, C₂².M₄(2), C₂×C₂²⋊C₈, C₂⁴.₄C₄, C₂×C₂².D₄, C₂³⋊C₈⋊C₂
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₈○D₄, (C₂²×C₈)⋊C₂, C₂×C₂³⋊C₄, M₄(2).₈C₂², C₂³⋊C₈⋊C₂

Smallest permutation representation of C₂³⋊C₈⋊C₂
►On 32 points

Generators in S₃₂

(1 5)(2 32)(3 29)(6 28)(7 25)(9 13)(11 18)(12 23)(15 22)(16 19)(20 24)(27 31)

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+				+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	C₈○D₄	C₂³⋊C₄	M₄(2).₈C₂²
kernel	C₂³⋊C₈⋊C₂	C₂³⋊C₈	C₂².M₄(2)	C₂×C₂²⋊C₈	C₂⁴.₄C₄	C₂×C₂².D₄	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₂²×D₄	C₂²×C₄	C₂²	C₂²	C₂
# reps	1	2	2	1	1	1	4	2	2	4	8	2	2

Matrix representation of C₂³⋊C₈⋊C₂ ►in GL₆(𝔽₁₇)

16	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	0	0	0	16

1	0	0	0	0	0
0	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	16

1	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	0	0	16	0
0	0	0	0	0	16

2	0	0	0	0	0
0	2	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	1	0	0	0

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

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C₂³⋊C₈⋊C₂ in GAP, Magma, Sage, TeX

C_2^3\rtimes C_8\rtimes C_2

% in TeX

G:=Group("C2^3:C8:C2");

// GroupNames label

G:=SmallGroup(128,200);

// by ID

G=gap.SmallGroup(128,200);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,1123,851,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,e*a*e=a*d^4,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;

// generators/relations

G = C23⋊C8⋊C2 order 128 = 27

15th semidirect product of C23⋊C8 and C2 acting faithfully

G = C₂³⋊C₈⋊C₂ order 128 = 2⁷

15^th semidirect product of C₂³⋊C₈ and C₂ acting faithfully