(C2^3xC4).C4 - GroupNames

G = (C₂³xC₄).C₄ order 128 = 2⁷

20^th non-split extension by C₂³xC₄ of C₄ acting faithfully

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

Aliases: C₄.₅₁C₂²wrC₂, (C₂xD₄).₂₆₂D₄, C₂⁴.₂₈(C₂xC₄), (C₂³xC₄).₂₀C₄, (C₂xQ₈).₂₀₅D₄, (C₂²xD₄).₂₈C₄, (C₂²xC₄).₂₆₁D₄, C₂⁴.₄C₄:₂₄C₂, C₂.₁₀(C₂⁴:₃C₄), C₂³.₃₀(C₂²:C₄), C₂³.₁₈₄(C₂²xC₄), (C₂³xC₄).₂₃₀C₂², (C₂²xC₄).₆₅₅C₂³, (C₂²xD₄).₄₅₁C₂², (C₂²xQ₈).₃₇₉C₂², (C₂xM₄(2)).₁₄₈C₂², C₂.₂₂(M₄(2).₈C₂²), (C₂xC₄).₂₂₆(C₂xD₄), (C₂xC₄.D₄):₁₄C₂, (C₂²xC₄oD₄).₃C₂, (C₂xC₄.₁₀D₄):₁₄C₂, (C₂xC₄).₄₃(C₂²:C₄), (C₂²xC₄).₄₄₁(C₂xC₄), C₂².₃₂(C₂xC₂²:C₄), SmallGroup(128,517)

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

Derived series

C₁ — C₂³ — (C₂³xC₄).C₄

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂³xC₄ — C₂²xC₄oD₄ — (C₂³xC₄).C₄

Lower central

C₁ — C₂ — C₂³ — (C₂³xC₄).C₄

Upper central

C₁ — C₂² — C₂³xC₄ — (C₂³xC₄).C₄

Jennings

C₁ — C₂ — C₂ — C₂²xC₄ — (C₂³xC₄).C₄

Generators and relations for (C₂³xC₄).C₄
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e⁴=d², ab=ba, ac=ca, ad=da, eae^-1=ad², bc=cb, ede^-1=bd=db, be=eb, cd=dc, ece^-1=acd² >

Subgroups: 580 in 292 conjugacy classes, 68 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂xC₈, M₄(2), C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₄oD₄, C₂⁴, C₂⁴, C₂²:C₈, C₄.D₄, C₄.₁₀D₄, C₂xM₄(2), C₂³xC₄, C₂³xC₄, C₂²xD₄, C₂²xD₄, C₂²xQ₈, C₂xC₄oD₄, C₂⁴.₄C₄, C₂xC₄.D₄, C₂xC₄.₁₀D₄, C₂²xC₄oD₄, (C₂³xC₄).C₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂xC₂²:C₄, C₂²wrC₂, C₂⁴:₃C₄, M₄(2).₈C₂², (C₂³xC₄).C₄

Smallest permutation representation of (C₂³xC₄).C₄
►On 32 points

Generators in S₃₂

(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)

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

(2 6)(3 7)(11 15)(12 16)(17 21)(20 24)(25 29)(28 32)

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

(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)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	4
type	+	+	+	+	+			+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₄	M₄(2).₈C₂²
kernel	(C₂³xC₄).C₄	C₂⁴.₄C₄	C₂xC₄.D₄	C₂xC₄.₁₀D₄	C₂²xC₄oD₄	C₂³xC₄	C₂²xD₄	C₂²xC₄	C₂xD₄	C₂xQ₈	C₂
# reps	1	2	2	2	1	4	4	4	4	4	4

Matrix representation of (C₂³xC₄).C₄ ►in GL₆(F₁₇)

16	0	0	0	0	0
0	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	16

16	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	0
0	0	0	0	0	16

0	1	0	0	0	0
1	0	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	16	0	0	0	0
16	0	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	13	0
0	0	0	0	0	13

4	0	0	0	0	0
0	13	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	16	0	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C₂³xC₄).C₄ in GAP, Magma, Sage, TeX

(C_2^3\times C_4).C_4

% in TeX

G:=Group("(C2^3xC4).C4");

// GroupNames label

G:=SmallGroup(128,517);

// by ID

G=gap.SmallGroup(128,517);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,2028,124]);

// Polycyclic

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

// generators/relations

G = (C23xC4).C4 order 128 = 27

20th non-split extension by C23xC4 of C4 acting faithfully

G = (C₂³xC₄).C₄ order 128 = 2⁷

20^th non-split extension by C₂³xC₄ of C₄ acting faithfully