C2^4.(C2xC4) - GroupNames

G = C₂⁴.(C₂×C₄) order 128 = 2⁷

3^rd non-split extension by C₂⁴ of C₂×C₄ acting faithfully

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

Aliases: C₂³⋊C₈⋊₁₇C₂, C₂⁴.₃(C₂×C₄), C₄.₉(C₂³⋊C₄), (C₂²×D₄).₆C₄, (C₂²×C₄).₇₂₆D₄, C₂⁴.₄C₄⋊₁₈C₂, C₂².₁₂(C₈○D₄), C₂²⋊C₈.₁₅₇C₂², C₂³.₄₁(C₂²⋊C₄), C₂³.₁₇₂(C₂²×C₄), (C₂³×C₄).₂₀₁C₂², (C₂²×C₄).₄₃₅C₂³, C₂².₁(C₄.D₄), (C₂×C₄⋊C₄).₁₃C₄, (C₂×C₂²⋊C₈)⋊₄C₂, (C₂×C₄⋊D₄).₁C₂, C₂.₁₀(C₂×C₂³⋊C₄), C₂.₈(C₂×C₄.D₄), (C₂×C₄).₁₁₃₂(C₂×D₄), (C₂²×C₄).₁₂(C₂×C₄), (C₂×C₄).₇₁(C₂²⋊C₄), C₂.₉((C₂²×C₈)⋊C₂), (C₂×C₂²⋊C₄).₈₈C₂², C₂².₁₅₃(C₂×C₂²⋊C₄), SmallGroup(128,203)

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

Derived series

C₁ — C₂³ — C₂⁴.(C₂×C₄)

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₄⋊D₄ — C₂⁴.(C₂×C₄)

Lower central

C₁ — C₂ — C₂³ — C₂⁴.(C₂×C₄)

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.(C₂×C₄)

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.(C₂×C₄)

Generators and relations for C₂⁴.(C₂×C₄)
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e²=f⁴=d, ab=ba, ac=ca, eae^-1=ad=da, faf^-1=abc, fbf^-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 412 in 161 conjugacy classes, 48 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂²×C₈, C₂×M₄(2), C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³⋊C₈, C₂×C₂²⋊C₈, C₂⁴.₄C₄, C₂×C₄⋊D₄, C₂⁴.(C₂×C₄)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂³⋊C₄, C₄.D₄, C₂×C₂²⋊C₄, C₈○D₄, (C₂²×C₈)⋊C₂, C₂×C₂³⋊C₄, C₂×C₄.D₄, C₂⁴.(C₂×C₄)

Smallest permutation representation of C₂⁴.(C₂×C₄)
►On 32 points

Generators in S₃₂

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

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

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

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

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

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+			+		+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	D₄	C₈○D₄	C₂³⋊C₄	C₄.D₄
kernel	C₂⁴.(C₂×C₄)	C₂³⋊C₈	C₂×C₂²⋊C₈	C₂⁴.₄C₄	C₂×C₄⋊D₄	C₂×C₄⋊C₄	C₂²×D₄	C₂²×C₄	C₂²	C₄	C₂²
# reps	1	4	1	1	1	2	6	4	8	2	2

Matrix representation of C₂⁴.(C₂×C₄) ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	1	0	0	0	0
0	0	0	16	0	0
0	0	16	0	0	0
0	0	10	0	16	2
0	0	5	5	0	1

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	10	7	16	0
0	0	0	7	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

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	16	0	0	0	0
1	0	0	0	0	0
0	0	0	16	0	0
0	0	1	0	0	0
0	0	0	0	16	2
0	0	0	0	16	1

0	2	0	0	0	0
15	0	0	0	0	0
0	0	12	5	1	0
0	0	12	12	1	15
0	0	0	0	0	10
0	0	0	0	12	10

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

C₂⁴.(C₂×C₄) in GAP, Magma, Sage, TeX

C_2^4.(C_2\times C_4)

% in TeX

G:=Group("C2^4.(C2xC4)");

// GroupNames label

G:=SmallGroup(128,203);

// by ID

G=gap.SmallGroup(128,203);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,352,1123,851,172]);

// Polycyclic

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

// generators/relations

G = C24.(C2×C4) order 128 = 27

3rd non-split extension by C24 of C2×C4 acting faithfully

G = C₂⁴.(C₂×C₄) order 128 = 2⁷

3^rd non-split extension by C₂⁴ of C₂×C₄ acting faithfully