C4^2.297C2^3 - GroupNames

G = C₄².₂₉₇C₂³ order 128 = 2⁷

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

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₄².₂₉₇C₂³, C₄.₁₇₀2₊¹⁺⁴, (C₈xD₄):₄₆C₂, C₈:₆D₄:₄₁C₂, C₈:₉D₄:₄₁C₂, C₄:C₈:₅₂C₂², (C₄xC₈):₆₁C₂², C₂²wrC₂.₅C₄, C₄:D₄.₂₄C₄, C₂⁴.₈₇(C₂xC₄), C₈:C₄:₃₁C₂², C₂²:Q₈.₂₄C₄, C₂²:C₈:₈₀C₂², (C₂xC₈).₄₃₄C₂³, (C₂xC₄).₆₇₃C₂⁴, (C₂²xC₈):₅₆C₂², C₂².₄(C₈oD₄), C₂⁴.₄C₄:₃₇C₂, (C₄xD₄).₃₀₀C₂², C₂².D₄.₈C₄, C₂.₂₉(Q₈oM₄(2)), (C₂xM₄(2)):₄₇C₂², (C₂³xC₄).₅₃₂C₂², C₂³.₁₅₀(C₂²xC₄), C₂².₁₉₇(C₂³xC₄), C₄²:C₂.₈₅C₂², (C₂²xC₄).₁₂₈₂C₂³, C₄².₇C₂²:₂₆C₂, C₂².₁₉C₂⁴.₁₂C₂, C₂.₄₇(C₂².₁₁C₂⁴), C₂.₂₈(C₂xC₈oD₄), C₄:C₄.₁₆₇(C₂xC₄), (C₂xC₂²:C₈):₄₇C₂, (C₂xD₄).₁₈₂(C₂xC₄), C₂²:C₄.₁₉(C₂xC₄), (C₂xC₄).₇₉(C₂²xC₄), (C₂xQ₈).₁₂₂(C₂xC₄), (C₂²xC₈):C₂:₃₂C₂, (C₂²xC₄).₃₅₄(C₂xC₄), (C₂xC₄oD₄).₉₃C₂², SmallGroup(128,1708)

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

Derived series

C₁ — C₂² — C₄².₂₉₇C₂³

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂³xC₄ — C₂².₁₉C₂⁴ — C₄².₂₉₇C₂³

Lower central

C₁ — C₂² — C₄².₂₉₇C₂³

Upper central

C₁ — C₂xC₄ — C₄².₂₉₇C₂³

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₄².₂₉₇C₂³

Generators and relations for C₄².₂₉₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=e²=1, c²=b, ab=ba, cac^-1=dad=a^-1, eae=ab², bc=cb, bd=db, be=eb, dcd=ece=a²c, ede=b²d >

Subgroups: 348 in 206 conjugacy classes, 126 normal (40 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²:C₄, C₄:C₄, C₄:C₄, C₂xC₈, C₂xC₈, M₄(2), C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₄xC₈, C₈:C₄, C₂²:C₈, C₄:C₈, C₄²:C₂, C₄xD₄, C₂²wrC₂, C₄:D₄, C₂²:Q₈, C₂².D₄, C₂²xC₈, C₂²xC₈, C₂xM₄(2), C₂xM₄(2), C₂³xC₄, C₂xC₄oD₄, C₂xC₂²:C₈, C₂⁴.₄C₄, (C₂²xC₈):C₂, C₄².₇C₂², C₈xD₄, C₈:₉D₄, C₈:₆D₄, C₂².₁₉C₂⁴, C₄².₂₉₇C₂³
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, C₂³, C₂²xC₄, C₂⁴, C₈oD₄, C₂³xC₄, 2₊¹⁺⁴, C₂².₁₁C₂⁴, C₂xC₈oD₄, Q₈oM₄(2), C₄².₂₉₇C₂³

Smallest permutation representation of C₄².₂₉₇C₂³
►On 32 points

Generators in S₃₂

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

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

(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 20)(11 22)(13 24)(15 18)(25 29)(27 31)

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	···	4N	8A	···	8H	8I	···	8T
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	1	1	2	2	4	4	4	4	1	1	1	1	2	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+	+						+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₈oD₄	2₊¹⁺⁴	Q₈oM₄(2)
kernel	C₄².₂₉₇C₂³	C₂xC₂²:C₈	C₂⁴.₄C₄	(C₂²xC₈):C₂	C₄².₇C₂²	C₈xD₄	C₈:₉D₄	C₈:₆D₄	C₂².₁₉C₂⁴	C₂²wrC₂	C₄:D₄	C₂²:Q₈	C₂².D₄	C₂²	C₄	C₂
# reps	1	1	1	2	2	2	4	2	1	4	4	4	4	8	2	2

Matrix representation of C₄².₂₉₇C₂³ ►in GL₆(F₁₇)

16	15	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	1	1	1	1
0	0	0	15	15	16

13	0	0	0	0	0
0	13	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

8	0	0	0	0	0
0	8	0	0	0	0
0	0	0	0	1	0
0	0	16	16	16	16
0	0	1	0	0	0
0	0	0	0	0	1

16	15	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	1	1	1	1
0	0	15	15	0	16

16	0	0	0	0	0
1	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	15	15	0	16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,16,1,0,0,0,1,0,1,15,0,0,0,0,1,15,0,0,0,0,1,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,0,0,16,0,1],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,1,1,15,0,0,1,0,1,15,0,0,0,0,1,0,0,0,0,0,1,16],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,15,0,0,0,1,0,15,0,0,0,0,16,0,0,0,0,0,0,16] >;

C₄².₂₉₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{297}C_2^3

% in TeX

G:=Group("C4^2.297C2^3");

// GroupNames label

G:=SmallGroup(128,1708);

// by ID

G=gap.SmallGroup(128,1708);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,124]);

// Polycyclic

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

// generators/relations

G = C42.297C23 order 128 = 27

158th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₉₇C₂³ order 128 = 2⁷

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