C2xD8:C2^2 - GroupNames

G = C₂×D₈⋊C₂² order 128 = 2⁷

Direct product of C₂ and D₈⋊C₂²

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

Aliases: C₂×D₈⋊C₂², D₈⋊₄C₂³, C₈.₂C₂⁴, C₄.₇C₂⁵, Q₁₆⋊₄C₂³, D₄.₄C₂⁴, Q₈.₄C₂⁴, SD₁₆⋊₃C₂³, C₂⁴.₁₃₁D₄, M₄(2)⋊₆C₂³, (C₂×C₈)⋊₅C₂³, C₄○D₈⋊₈C₂², C₄○D₄⋊₆C₂³, (C₂×D₈)⋊₅₅C₂², (C₂×D₄)⋊₂₂C₂³, (C₂×Q₈)⋊₂₂C₂³, C₂.₄₂(D₄×C₂³), C₄ ○(D₈⋊C₂²), C₈⋊C₂²⋊₁₇C₂², (C₂×C₄).₆₁₃C₂⁴, (C₂²×C₈)⋊₂₇C₂², (C₂×Q₁₆)⋊₅₉C₂², C₂³.₃₅₇(C₂×D₄), C₄.₁₇₄(C₂²×D₄), (C₂²×C₄).₅₃₈D₄, (C₂×SD₁₆)⋊₆₁C₂², (C₂²×D₄)⋊₆₆C₂², C₈.C₂²⋊₁₇C₂², (C₂²×M₄(2))⋊₈C₂, (C₂²×Q₈)⋊₇₀C₂², C₂².₅₄(C₂²×D₄), (C₂×M₄(2))⋊₅₈C₂², (C₂³×C₄).₆₂₄C₂², (C₂²×C₄).₁₂₂₄C₂³, C₄ ○(C₂×C₈⋊C₂²), C₄ ○(C₂×C₈.C₂²), (C₂×C₄○D₈)⋊₂₉C₂, (C₂×C₄)○(C₈⋊C₂²), (C₂×C₈⋊C₂²)⋊₃₇C₂, (C₂×C₄).₆₆₉(C₂×D₄), (C₂×C₄)○(C₈.C₂²), (C₂×C₄○D₄)⋊₇₈C₂², (C₂²×C₄○D₄)⋊₂₇C₂, (C₂×C₈.C₂²)⋊₃₇C₂, (C₂×C₄)○(C₂×C₈.C₂²), SmallGroup(128,2312)

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

Derived series

C₁ — C₄ — C₂×D₈⋊C₂²

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂×D₈⋊C₂²

Lower central

C₁ — C₂ — C₄ — C₂×D₈⋊C₂²

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂×D₈⋊C₂²

Jennings

C₁ — C₂ — C₂ — C₄ — C₂×D₈⋊C₂²

Generators and relations for C₂×D₈⋊C₂²
G = < a,b,c,d,e | a²=b⁸=c²=d²=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, dbd=b⁵, be=eb, dcd=ece=b⁴c, de=ed >

Subgroups: 1148 in 742 conjugacy classes, 428 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂⁴, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₈.C₂², C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂²×M₄(2), C₂×C₄○D₈, C₂×C₈⋊C₂², C₂×C₈.C₂², D₈⋊C₂², C₂²×C₄○D₄, C₂×D₈⋊C₂²
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, C₂⁵, D₈⋊C₂², D₄×C₂³, C₂×D₈⋊C₂²

Smallest permutation representation of C₂×D₈⋊C₂²
►On 32 points

Generators in S₃₂

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

(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)

(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2Q	4A	4B	4C	4D	4E	···	4J	4K	···	4R	8A	···	8H
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	2	2	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₈⋊C₂²
kernel	C₂×D₈⋊C₂²	C₂²×M₄(2)	C₂×C₄○D₈	C₂×C₈⋊C₂²	C₂×C₈.C₂²	D₈⋊C₂²	C₂²×C₄○D₄	C₂²×C₄	C₂⁴	C₂
# reps	1	1	4	4	4	16	2	7	1	4

Matrix representation of C₂×D₈⋊C₂² ►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	16	0	0	0	0
1	0	0	0	0	0
0	0	0	0	4	0
0	0	0	0	13	13
0	0	4	8	0	0
0	0	0	13	0	0

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

16	0	0	0	0	0
0	16	0	0	0	0
0	0	4	8	0	0
0	0	13	13	0	0
0	0	0	0	4	8
0	0	0	0	13	13

16	0	0	0	0	0
0	16	0	0	0	0
0	0	4	8	0	0
0	0	13	13	0	0
0	0	0	0	13	9
0	0	0	0	4	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],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,8,13,0,0,4,13,0,0,0,0,0,13,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,13,0,0,0,0,8,13,0,0,0,0,0,0,4,13,0,0,0,0,8,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,13,0,0,0,0,8,13,0,0,0,0,0,0,13,4,0,0,0,0,9,4] >;

C₂×D₈⋊C₂² in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes C_2^2

% in TeX

G:=Group("C2xD8:C2^2");

// GroupNames label

G:=SmallGroup(128,2312);

// by ID

G=gap.SmallGroup(128,2312);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,1430,248,4037,2028,124]);

// Polycyclic

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

// generators/relations

G = C2×D8⋊C22 order 128 = 27

Direct product of C2 and D8⋊C22

G = C₂×D₈⋊C₂² order 128 = 2⁷

Direct product of C₂ and D₈⋊C₂²