C4^2.257C2^3 - GroupNames

G = C₄².₂₅₇C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₂₅₇C₂³, C₄⋊C₈⋊₄₈C₂², C₂⁴.₇₅(C₂×C₄), (C₂²×C₄).₆₃Q₈, C₄.₆₀(C₂²×Q₈), C₂³.₃₁(C₄⋊C₄), (C₂×C₈).₃₉₅C₂³, (C₂×C₄).₆₃₄C₂⁴, C₄².₂₀₁(C₂×C₄), (C₂²×C₄).₄₁₄D₄, C₄.₁₈₆(C₂²×D₄), C₄⋊M₄(2)⋊₃₀C₂, C₄²⋊C₂.₂₇C₄, C₂.₈(Q₈○M₄(2)), (C₂×C₄²).₇₅₂C₂², (C₂²×C₈).₄₂₇C₂², (C₂³×C₄).₅₁₇C₂², C₂³.₁₃₈(C₂²×C₄), C₂².₁₆₃(C₂³×C₄), (C₂²×C₄).₁₅₀₂C₂³, C₄².₆C₂²⋊₂₆C₂, (C₂²×M₄(2)).₂₉C₂, C₄²⋊C₂.₂₈₃C₂², (C₂×M₄(2)).₃₃₇C₂², C₄.₆₂(C₂×C₄⋊C₄), (C₂×C₄⋊C₄).₆₆C₄, C₄⋊C₄.₂₁₃(C₂×C₄), (C₂×C₄).₆₀(C₄⋊C₄), (C₂×C₄).₈₄₀(C₂×D₄), C₂².₃₄(C₂×C₄⋊C₄), C₂.₂₀(C₂²×C₄⋊C₄), (C₂×C₄).₂₃₆(C₂×Q₈), (C₂×C₂²⋊C₄).₄₃C₄, C₂²⋊C₄.₆₄(C₂×C₄), (C₂×C₄).₂₄₈(C₂²×C₄), (C₂²×C₄).₃₂₈(C₂×C₄), (C₂×C₄²⋊C₂).₅₂C₂, SmallGroup(128,1637)

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

Derived series

C₁ — C₂² — C₄².₂₅₇C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₄²⋊C₂ — C₄².₂₅₇C₂³

Lower central

C₁ — C₂² — C₄².₂₅₇C₂³

Upper central

C₁ — C₂×C₄ — C₄².₂₅₇C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₅₇C₂³

Generators and relations for C₄².₂₅₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=e²=1, c²=b, ab=ba, cac^-1=a^-1, dad=ab², ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b²c, de=ed >

Subgroups: 332 in 242 conjugacy classes, 172 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂⁴, C₄⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₄⋊M₄(2), C₄².₆C₂², C₂×C₄²⋊C₂, C₂²×M₄(2), C₄².₂₅₇C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂²×C₄⋊C₄, Q₈○M₄(2), C₄².₂₅₇C₂³

Smallest permutation representation of C₄².₂₅₇C₂³
►On 32 points

Generators in S₃₂

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

(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)

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

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

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

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

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	1	2	2	4
type	+	+	+	+	+				+	-
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	Q₈○M₄(2)
kernel	C₄².₂₅₇C₂³	C₄⋊M₄(2)	C₄².₆C₂²	C₂×C₄²⋊C₂	C₂²×M₄(2)	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₄²⋊C₂	C₂²×C₄	C₂²×C₄	C₂
# reps	1	4	8	1	2	4	4	8	4	4	4

Matrix representation of C₄².₂₅₇C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
1	13	0	0	0	0
0	0	5	15	0	0
0	0	12	12	0	0
0	0	4	0	4	15
0	0	1	13	16	13

16	0	0	0	0	0
0	16	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	13	0
0	0	0	0	0	13

16	8	0	0	0	0
4	1	0	0	0	0
0	0	0	0	1	0
0	0	15	0	9	1
0	0	13	0	0	0
0	0	2	13	2	0

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	5	16	0	0
0	0	0	0	1	0
0	0	4	0	4	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	4	0	0	16

G:=sub<GL(6,GF(17))| [4,1,0,0,0,0,0,13,0,0,0,0,0,0,5,12,4,1,0,0,15,12,0,13,0,0,0,0,4,16,0,0,0,0,15,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,4,0,0,0,0,8,1,0,0,0,0,0,0,0,15,13,2,0,0,0,0,0,13,0,0,1,9,0,2,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,5,0,4,0,0,0,16,0,0,0,0,0,0,1,4,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C₄².₂₅₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{257}C_2^3

% in TeX

G:=Group("C4^2.257C2^3");

// GroupNames label

G:=SmallGroup(128,1637);

// by ID

G=gap.SmallGroup(128,1637);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,2019,124]);

// 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=a^-1,d*a*d=a*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,d*e=e*d>;

// generators/relations

G = C42.257C23 order 128 = 27

118th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₅₇C₂³ order 128 = 2⁷

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