C2xD4oD8 - GroupNames

G = C₂×D₄○D₈ order 128 = 2⁷

Direct product of C₂ and D₄○D₈

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

Aliases: C₂×D₄○D₈, D₈⋊₈C₂³, C₈.₃C₂⁴, C₄.₈C₂⁵, Q₁₆⋊₈C₂³, D₄.₅C₂⁴, Q₈.₅C₂⁴, SD₁₆⋊₄C₂³, M₄(2)⋊₇C₂³, 2₊¹⁺⁴⋊₈C₂², D₄ ○(C₂×D₈), D₈ ○(C₂×D₄), Q₁₆ ○(C₂×Q₈), Q₈ ○(C₂×Q₁₆), (C₂×C₈)⋊₆C₂³, C₄○D₄.₃₈D₄, D₄.₆₁(C₂×D₄), C₄○D₈⋊₉C₂², C₄○D₄⋊₁C₂³, Q₈.₆₃(C₂×D₄), (C₂×D₄).₃₅₇D₄, C₈○D₄⋊₁₃C₂², (C₂²×D₈)⋊₂₃C₂, (C₂×D₄)⋊₁₁C₂³, (C₂×D₈)⋊₅₆C₂², (C₂×Q₈).₂₇₆D₄, C₂.₄₃(D₄×C₂³), C₈⋊C₂²⋊₁₂C₂², (C₂×C₄).₆₁₄C₂⁴, (C₂²×C₈)⋊₂₈C₂², (C₂×Q₁₆)⋊₆₄C₂², C₄.₁₂₅(C₂²×D₄), C₂³.₄₈₅(C₂×D₄), (C₂×SD₁₆)⋊₆₂C₂², (C₂²×D₄)⋊₅₀C₂², (C₂×Q₈).₄₇₅C₂³, C₂².₁₇(C₂²×D₄), (C₂×M₄(2))⋊₅₉C₂², (C₂×2₊¹⁺⁴)⋊₁₃C₂, (C₂²×C₄).₁₂₂₅C₂³, (C₂×Q₈)○(C₂×Q₁₆), (C₂×C₄○D₈)⋊₃₀C₂, (C₂×C₈○D₄)⋊₁₀C₂, (C₂×C₈⋊C₂²)⋊₃₄C₂, (C₂×C₄).₁₁₁₃(C₂×D₄), (C₂×C₄○D₄)⋊₅₆C₂², SmallGroup(128,2313)

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

Derived series

C₁ — C₄ — C₂×D₄○D₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×2₊¹⁺⁴ — C₂×D₄○D₈

Lower central

C₁ — C₂ — C₄ — C₂×D₄○D₈

Upper central

C₁ — C₂² — C₂×C₄○D₄ — C₂×D₄○D₈

Jennings

C₁ — C₂ — C₂ — C₄ — C₂×D₄○D₈

Generators and relations for C₂×D₄○D₈
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁴=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=b²d³ >

Subgroups: 1348 in 756 conjugacy classes, 428 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂²×C₈, C₂×M₄(2), C₈○D₄, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, C₂×C₈○D₄, C₂²×D₈, C₂×C₄○D₈, C₂×C₈⋊C₂², D₄○D₈, C₂×2₊¹⁺⁴, C₂×D₄○D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, C₂⁵, D₄○D₈, D₄×C₂³, C₂×D₄○D₈

Smallest permutation representation of C₂×D₄○D₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	4
type	+	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₄○D₈
kernel	C₂×D₄○D₈	C₂×C₈○D₄	C₂²×D₈	C₂×C₄○D₈	C₂×C₈⋊C₂²	D₄○D₈	C₂×2₊¹⁺⁴	C₂×D₄	C₂×Q₈	C₄○D₄	C₂
# reps	1	1	3	3	6	16	2	3	1	4	4

Matrix representation of C₂×D₄○D₈ ►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

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

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

0	1	0	0	0	0
16	0	0	0	0	0
0	0	3	3	0	0
0	0	14	3	0	0
0	0	14	14	0	11
0	0	3	0	3	6

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

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,0,0,0,16,0,16,1,0,0,0,0,16,1,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,16,0,0,1,1,0,16,0,0,2,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,3,14,14,3,0,0,3,3,14,0,0,0,0,0,0,3,0,0,0,0,11,6],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,1,0,0,0,16,1,0,0,0,0,15,0,0,1] >;

C₂×D₄○D₈ in GAP, Magma, Sage, TeX

C_2\times D_4\circ D_8

% in TeX

G:=Group("C2xD4oD8");

// GroupNames label

G:=SmallGroup(128,2313);

// by ID

G=gap.SmallGroup(128,2313);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,521,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;

// generators/relations

G = C2×D4○D8 order 128 = 27

Direct product of C2 and D4○D8

G = C₂×D₄○D₈ order 128 = 2⁷

Direct product of C₂ and D₄○D₈