C2xM4(2).8C2^2 - GroupNames

G = C₂×M₄(2).₈C₂² order 128 = 2⁷

Direct product of C₂ and M₄(2).₈C₂²

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

Aliases: C₂×M₄(2).₈C₂², M₄(2).₂₃C₂³, (C₂×C₄).₃C₂⁴, C₂⁴.₃₅(C₂×C₄), (C₂³×C₄).₂₄C₄, (C₂²×D₄).₃₇C₄, (C₂²×C₄).₇₇₇D₄, C₄.₁₃₃(C₂²×D₄), (C₂×D₄).₃₅₄C₂³, C₄.D₄⋊₁₈C₂², C₂².₁₆(C₂³×C₄), C₂³.₆₀(C₂²×C₄), (C₂×Q₈).₃₂₇C₂³, C₄.₁₀D₄⋊₁₉C₂², (C₂²×M₄(2))⋊₁₇C₂, (C₂×M₄(2))⋊₆₇C₂², (C₂³×C₄).₅₁₁C₂², (C₂²×C₄).₈₉₇C₂³, C₂³.₁₂₉(C₂²⋊C₄), (C₂²×D₄).₅₄₉C₂², (C₂²×Q₈).₄₅₃C₂², C₄ ○(M₄(2).₈C₂²), C₄ ○(C₂×C₄.D₄), C₄ ○(C₂×C₄.₁₀D₄), (C₂×C₄○D₄).₂₄C₄, (C₂×C₄)○(C₄.D₄), (C₂×D₄).₂₂₃(C₂×C₄), (C₂×C₄)○(C₄.₁₀D₄), (C₂×C₄.D₄)⋊₂₉C₂, (C₂×C₄).₁₄₀₁(C₂×D₄), C₄.₁₂₀(C₂×C₂²⋊C₄), (C₂²×C₄).₈₈(C₂×C₄), (C₂×Q₈).₂₀₁(C₂×C₄), (C₂×C₄.₁₀D₄)⋊₂₉C₂, (C₂×C₄).₁₀₈(C₂²×C₄), (C₂²×C₄○D₄).₁₇C₂, C₂.₃₀(C₂²×C₂²⋊C₄), C₂².₁₉(C₂×C₂²⋊C₄), (C₂×C₄).₂₈₁(C₂²⋊C₄), (C₂×C₄○D₄).₂₇₁C₂², (C₂×C₄)○(C₂×C₄.D₄), SmallGroup(128,1619)

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

Derived series

C₁ — C₂² — C₂×M₄(2).₈C₂²

Chief series

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

Lower central

C₁ — C₂ — C₂² — C₂×M₄(2).₈C₂²

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂×M₄(2).₈C₂²

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×M₄(2).₈C₂²

Generators and relations for C₂×M₄(2).₈C₂²
G = < a,b,c,d,e | a²=b⁸=c²=d²=1, e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe^-1=b⁵, dbd=bc, cd=dc, ece^-1=b⁴c, ede^-1=b⁴cd >

Subgroups: 636 in 378 conjugacy classes, 172 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₄.D₄, C₄.₁₀D₄, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄.D₄, C₂×C₄.₁₀D₄, M₄(2).₈C₂², C₂²×M₄(2), C₂²×C₄○D₄, C₂×M₄(2).₈C₂²
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, M₄(2).₈C₂², C₂²×C₂²⋊C₄, C₂×M₄(2).₈C₂²

Smallest permutation representation of C₂×M₄(2).₈C₂²
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+	+				+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	M₄(2).₈C₂²
kernel	C₂×M₄(2).₈C₂²	C₂×C₄.D₄	C₂×C₄.₁₀D₄	M₄(2).₈C₂²	C₂²×M₄(2)	C₂²×C₄○D₄	C₂³×C₄	C₂²×D₄	C₂×C₄○D₄	C₂²×C₄	C₂
# reps	1	2	2	8	2	1	4	4	8	8	4

Matrix representation of C₂×M₄(2).₈C₂² ►in GL₆(𝔽₁₇)

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

0	13	0	0	0	0
13	0	0	0	0	0
0	0	0	0	13	0
0	0	4	1	16	8
0	0	1	0	0	0
0	0	6	11	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	1	0
0	0	1	13	0	1

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	16	4	13	15
0	0	0	1	16	4

0	13	0	0	0	0
13	0	0	0	0	0
0	0	0	0	16	0
0	0	1	13	4	2
0	0	4	0	0	0
0	0	11	11	0	4

G:=sub<GL(6,GF(17))| [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],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,4,1,6,0,0,0,1,0,11,0,0,13,16,0,0,0,0,0,8,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,16,0,13,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,16,0,4,1,0,0,0,0,13,16,0,0,0,0,15,4],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,1,4,11,0,0,0,13,0,11,0,0,16,4,0,0,0,0,0,2,0,4] >;

C₂×M₄(2).₈C₂² in GAP, Magma, Sage, TeX

C_2\times M_4(2)._8C_2^2

% in TeX

G:=Group("C2xM4(2).8C2^2");

// GroupNames label

G:=SmallGroup(128,1619);

// by ID

G=gap.SmallGroup(128,1619);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,2028,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^2=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^5,d*b*d=b*c,c*d=d*c,e*c*e^-1=b^4*c,e*d*e^-1=b^4*c*d>;

// generators/relations

G = C2×M4(2).8C22 order 128 = 27

Direct product of C2 and M4(2).8C22

G = C₂×M₄(2).₈C₂² order 128 = 2⁷

Direct product of C₂ and M₄(2).₈C₂²