C2^3.21C4^2 - GroupNames

G = C₂³.₂₁C₄² order 128 = 2⁷

3^rd non-split extension by C₂³ of C₄² acting via C₄²/C₂×C₄=C₂

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

Aliases: C₂³.₂₁C₄², C₂³.₅M₄(2), C₂²⋊C₈⋊₅C₄, C₂²⋊C₄⋊₂C₈, (C₂²×C₈)⋊₁C₄, C₂².₆(C₄×C₈), C₂³.₅(C₂×C₈), (C₂²×C₄).₄Q₈, C₂².₉(C₄⋊C₈), C₂⁴.₃₆(C₂×C₄), C₄.₃₂(C₂³⋊C₄), C₂³.₃₇(C₄⋊C₄), (C₂²×C₄).₆₃₇D₄, (C₂³×C₄).₂C₂², C₂².₅(C₈⋊C₄), C₂².₁₀(C₂²⋊C₈), C₂.₂(C₂³.₉D₄), C₂.₂(M₄(2)⋊₄C₄), C₂³.₁₃₅(C₂²⋊C₄), C₂.₁₁(C₂².₇C₄²), C₂².₂₃(C₂.C₄²), (C₂×C₄).₆₇(C₄⋊C₄), (C₂×C₂²⋊C₈).₂C₂, (C₄×C₂²⋊C₄).₁C₂, (C₂×C₂²⋊C₄).₁₄C₄, (C₂²×C₄).₄₂₄(C₂×C₄), (C₂×C₄).₃₇₁(C₂²⋊C₄), SmallGroup(128,14)

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

Derived series

C₁ — C₂² — C₂³.₂₁C₄²

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊C₈ — C₂³.₂₁C₄²

Lower central

C₁ — C₂ — C₂² — C₂³.₂₁C₄²

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂³.₂₁C₄²

Jennings

C₁ — C₂ — C₂² — C₂³×C₄ — C₂³.₂₁C₄²

Generators and relations for C₂³.₂₁C₄²
G = < a,b,c,d,e | a²=b²=c²=1, d⁴=e⁴=c, dad^-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=abd >

Subgroups: 240 in 114 conjugacy classes, 46 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂²×C₈, C₂²×C₈, C₂³×C₄, C₄×C₂²⋊C₄, C₂×C₂²⋊C₈, C₂³.₂₁C₄²
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂.C₄², C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₂³⋊C₄, C₄⋊C₈, C₂².₇C₄², C₂³.₉D₄, M₄(2)⋊₄C₄, C₂³.₂₁C₄²

Smallest permutation representation of C₂³.₂₁C₄²
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4J	4K	···	4R	8A	···	8P
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	4	···	4	8	···	8
size	1	1	1	1	2	···	2	1	1	1	1	2	···	2	4	···	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+					+	-		+
image	C₁	C₂	C₂	C₄	C₄	C₄	C₈	D₄	Q₈	M₄(2)	C₂³⋊C₄	M₄(2)⋊₄C₄
kernel	C₂³.₂₁C₄²	C₄×C₂²⋊C₄	C₂×C₂²⋊C₈	C₂²⋊C₈	C₂×C₂²⋊C₄	C₂²×C₈	C₂²⋊C₄	C₂²×C₄	C₂²×C₄	C₂³	C₄	C₂
# reps	1	1	2	4	4	4	16	3	1	4	2	2

Matrix representation of C₂³.₂₁C₄² ►in GL₆(𝔽₁₇)

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

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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

15	9	0	0	0	0
1	2	0	0	0	0
0	0	0	0	1	15
0	0	0	0	1	16
0	0	16	0	0	0
0	0	16	1	0	0

15	9	0	0	0	0
0	2	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	16	0	0	0
0	0	0	16	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,2,1],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,1,0,0,0,0,9,2,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,1,1,0,0,0,0,15,16,0,0],[15,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂³.₂₁C₄² in GAP, Magma, Sage, TeX

C_2^3._{21}C_4^2

% in TeX

G:=Group("C2^3.21C4^2");

// GroupNames label

G:=SmallGroup(128,14);

// by ID

G=gap.SmallGroup(128,14);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,570,248,172]);

// Polycyclic

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

// generators/relations

G = C23.21C42 order 128 = 27

3rd non-split extension by C23 of C42 acting via C42/C2×C4=C2

G = C₂³.₂₁C₄² order 128 = 2⁷

3^rd non-split extension by C₂³ of C₄² acting via C₄²/C₂×C₄=C₂