C2^3.8D8 - GroupNames

G = C₂³.₈D₈ order 128 = 2⁷

1^st non-split extension by C₂³ of D₈ acting via D₈/C₄=C₂²

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

Aliases: C₂³.₈D₈, C₂⁴.₄₇D₄, C₂³.₄Q₁₆, C₂³.₁₁SD₁₆, (C₂²×C₈)⋊₂C₄, (C₂×C₄).₃₃C₄², (C₂²×C₄).₁₈Q₈, C₄.₂₂(C₂³⋊C₄), C₂³.₃₈(C₄⋊C₄), C₂.C₄²⋊₂C₄, (C₂²×C₄).₁₇₈D₄, C₂².₈(C₂.D₈), C₂².₇(C₄.Q₈), C₂.₈(C₄.₉C₄²), C₂³.₇Q₈.₁C₂, C₂.₄(C₂².₄Q₁₆), (C₂³×C₄).₁₉₀C₂², C₂.₄(C₂³.₉D₄), C₂².₁₁(D₄⋊C₄), C₂³.₁₄₀(C₂²⋊C₄), C₂².₁₄(Q₈⋊C₄), C₂².₃₉(C₂.C₄²), (C₂×C₄).₁₆(C₄⋊C₄), (C₂×C₂²⋊C₈).₃C₂, (C₂²×C₄).₁₄₈(C₂×C₄), (C₂×C₄).₃₃₇(C₂²⋊C₄), SmallGroup(128,21)

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

Derived series

C₁ — C₂×C₄ — C₂³.₈D₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊C₈ — C₂³.₈D₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂³.₈D₈

Upper central

C₁ — C₂² — C₂³×C₄ — C₂³.₈D₈

Jennings

C₁ — C₂ — C₂² — C₂³×C₄ — C₂³.₈D₈

Generators and relations for C₂³.₈D₈
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=abc, ab=ba, eae^-1=ac=ca, ad=da, dbd^-1=ebe^-1=bc=cb, cd=dc, ce=ec, ede^-1=acd^-1 >

Subgroups: 280 in 114 conjugacy classes, 42 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂³×C₄, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂³.₈D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, D₈, SD₁₆, Q₁₆, C₂.C₄², C₂³⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄.₉C₄², C₂².₄Q₁₆, C₂³.₉D₄, C₂³.₈D₈

Smallest permutation representation of C₂³.₈D₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	4F	4G	···	4N	8A	···	8H
order	1	2	2	2	2	···	2	4	4	4	4	4	4	4	···	4	8	···	8
size	1	1	1	1	2	···	2	2	2	2	2	4	4	8	···	8	4	···	4

32 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+			+	-	+	+		-	+
image	C₁	C₂	C₂	C₄	C₄	D₄	Q₈	D₄	D₈	SD₁₆	Q₁₆	C₂³⋊C₄	C₄.₉C₄²
kernel	C₂³.₈D₈	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂.C₄²	C₂²×C₈	C₂²×C₄	C₂²×C₄	C₂⁴	C₂³	C₂³	C₂³	C₄	C₂
# reps	1	2	1	8	4	2	1	1	2	4	2	2	2

Matrix representation of C₂³.₈D₈ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	13	4	0	7
0	0	9	4	14	0
0	0	0	0	13	4
0	0	0	0	9	4

16	0	0	0	0	0
0	16	0	0	0	0
0	0	13	4	14	5
0	0	9	4	7	7
0	0	0	0	4	13
0	0	0	0	8	13

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

11	6	0	0	0	0
14	0	0	0	0	0
0	0	12	8	13	16
0	0	11	6	2	0
0	0	2	15	5	14
0	0	4	15	16	11

13	8	0	0	0	0
0	4	0	0	0	0
0	0	4	13	13	15
0	0	0	13	2	3
0	0	0	0	1	16
0	0	0	0	0	16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,9,0,0,0,0,4,4,0,0,0,0,0,14,13,9,0,0,7,0,4,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,9,0,0,0,0,4,4,0,0,0,0,14,7,4,8,0,0,5,7,13,13],[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],[11,14,0,0,0,0,6,0,0,0,0,0,0,0,12,11,2,4,0,0,8,6,15,15,0,0,13,2,5,16,0,0,16,0,14,11],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,13,13,0,0,0,0,13,2,1,0,0,0,15,3,16,16] >;

C₂³.₈D₈ in GAP, Magma, Sage, TeX

C_2^3._8D_8

% in TeX

G:=Group("C2^3.8D8");

// GroupNames label

G:=SmallGroup(128,21);

// by ID

G=gap.SmallGroup(128,21);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924,242]);

// Polycyclic

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

// generators/relations

G = C23.8D8 order 128 = 27

1st non-split extension by C23 of D8 acting via D8/C4=C22

G = C₂³.₈D₈ order 128 = 2⁷

1^st non-split extension by C₂³ of D₈ acting via D₈/C₄=C₂²