C2^3.8M4(2) - GroupNames

G = C₂³.₈M₄(2) order 128 = 2⁷

4^th non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₄=C₂²

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

Aliases: C₂³.₈M₄(2), (C₂×D₄)⋊₃C₈, C₂³⋊₁(C₂×C₈), C₄ ○₂(C₂³⋊C₈), C₂³⋊C₈⋊₁₈C₂, (C₂³×C₄).₆C₄, C₂⁴.₄₄(C₂×C₄), (C₂×C₄²).₁₅C₄, C₄.₄₄(C₂³⋊C₄), (C₂²×D₄).₁₈C₄, (C₂²×C₄).₆₅₀D₄, C₂².₉(C₂²×C₈), C₂².₁(C₂²⋊C₈), (C₂³×C₄).₁₇C₂², C₂²⋊C₈.₁₅₄C₂², C₂³.₉₀(C₂²⋊C₄), C₂³.₁₆₀(C₂²×C₄), (C₂²×C₄).₄₂₃C₂³, C₂².₁₂(C₂×M₄(2)), C₄ ○₂(C₂².M₄(2)), C₂².M₄(2)⋊₁₉C₂, C₂.₂(M₄(2).₈C₂²), (C₂×C₄)⋊₁(C₂×C₈), (C₂×C₄×D₄).₁C₂, (C₂×C₄)○(C₂³⋊C₈), (C₂×C₂²⋊C₈)⋊₂C₂, C₂.₇(C₂×C₂²⋊C₈), C₂.₃(C₂×C₂³⋊C₄), (C₂×C₄).₁₁₂₀(C₂×D₄), (C₂×C₂²⋊C₄).₁₇C₄, (C₂²×C₄).₄₀(C₂×C₄), (C₂×C₄⋊C₄).₇₃₃C₂², C₂².₉₁(C₂×C₂²⋊C₄), (C₂×C₄).₃₀₉(C₂²⋊C₄), (C₂×C₂²⋊C₄).₄₀₀C₂², (C₂×C₄)○(C₂².M₄(2)), SmallGroup(128,191)

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

Derived series

C₁ — C₂² — C₂³.₈M₄(2)

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₂³.₈M₄(2)

Lower central

C₁ — C₂ — C₂² — C₂³.₈M₄(2)

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂³.₈M₄(2)

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂³.₈M₄(2)

Generators and relations for C₂³.₈M₄(2)
G = < a,b,c,d,e | a²=b²=c²=d⁸=e²=1, ab=ba, dad^-1=eae=ac=ca, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede=bcd⁵ >

Subgroups: 348 in 166 conjugacy classes, 62 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂²⋊C₈, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂²×C₈, C₂³×C₄, C₂²×D₄, C₂³⋊C₈, C₂².M₄(2), C₂×C₂²⋊C₈, C₂×C₄×D₄, C₂³.₈M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂²⋊C₈, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂²×C₈, C₂×M₄(2), C₂×C₂²⋊C₈, C₂×C₂³⋊C₄, M₄(2).₈C₂², C₂³.₈M₄(2)

Smallest permutation representation of C₂³.₈M₄(2)
►On 32 points

Generators in S₃₂

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

(2 25)(4 27)(6 29)(8 31)(9 24)(11 18)(13 20)(15 22)

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+						+		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₈	D₄	M₄(2)	C₂³⋊C₄	M₄(2).₈C₂²
kernel	C₂³.₈M₄(2)	C₂³⋊C₈	C₂².M₄(2)	C₂×C₂²⋊C₈	C₂×C₄×D₄	C₂×C₄²	C₂×C₂²⋊C₄	C₂³×C₄	C₂²×D₄	C₂×D₄	C₂²×C₄	C₂³	C₄	C₂
# reps	1	2	2	2	1	2	2	2	2	16	4	4	2	2

Matrix representation of C₂³.₈M₄(2) ►in GL₆(𝔽₁₇)

1	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	11	16	0
0	0	11	0	0	1

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	10	11	16	0
0	0	6	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

4	13	0	0	0	0
5	13	0	0	0	0
0	0	6	7	0	15
0	0	10	11	15	0
0	0	0	4	6	7
0	0	0	0	10	11

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

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,11,0,0,0,1,11,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,10,6,0,0,0,1,11,7,0,0,0,0,16,0,0,0,0,0,0,16],[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],[4,5,0,0,0,0,13,13,0,0,0,0,0,0,6,10,0,0,0,0,7,11,4,0,0,0,0,15,6,10,0,0,15,0,7,11],[1,2,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0] >;

C₂³.₈M₄(2) in GAP, Magma, Sage, TeX

C_2^3._8M_4(2)

% in TeX

G:=Group("C2^3.8M4(2)");

// GroupNames label

G:=SmallGroup(128,191);

// by ID

G=gap.SmallGroup(128,191);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*c*d^5>;

// generators/relations

G = C23.8M4(2) order 128 = 27

4th non-split extension by C23 of M4(2) acting via M4(2)/C4=C22

G = C₂³.₈M₄(2) order 128 = 2⁷

4^th non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₄=C₂²