C2^3.30D8 - GroupNames

G = C₂³.₃₀D₈ order 128 = 2⁷

1^st non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

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

Aliases: C₂³.₃₀D₈, C₂⁴.₁₄₆D₄, C₂³.₁₅Q₁₆, C₂³.₂₈SD₁₆, C₂²⋊C₈⋊₆C₄, (C₂×C₄).₆C₄², C₂².₃₂C₄≀C₂, (C₂²×C₄).₂₂Q₈, C₂³.₄₂(C₄⋊C₄), C₂.C₄²⋊₉C₄, (C₂²×C₄).₆₃₈D₄, C₂².₁(C₄.Q₈), C₂.₈(C₄²⋊₆C₄), C₂².₁(C₂.D₈), C₂³.₇Q₈.₂C₂, C₂².₃₄(C₂³⋊C₄), C₂.₆(C₂².₄Q₁₆), (C₂³×C₄).₁₉₂C₂², C₂.₃(C₂².SD₁₆), C₂.₆(C₂³.₉D₄), C₂².₃₈(D₄⋊C₄), C₂³.₂₁₄(C₂²⋊C₄), C₂².₂₈(Q₈⋊C₄), C₂.₃(C₂³.₃₁D₄), C₂².₄₄(C₂.C₄²), (C₂×C₄⋊C₄)⋊₁C₄, (C₂×C₄).₇₀(C₄⋊C₄), (C₂×C₂²⋊C₈).₅C₂, (C₂²×C₄).₁₄₉(C₂×C₄), (C₂×C₄).₂₉₉(C₂²⋊C₄), (C₂×C₂.C₄²).₁₁C₂, SmallGroup(128,26)

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

Derived series

C₁ — C₂×C₄ — C₂³.₃₀D₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂³.₃₀D₈

Lower central

C₁ — C₂² — C₂×C₄ — C₂³.₃₀D₈

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂³.₃₀D₈

Jennings

C₁ — C₂ — C₂² — C₂³×C₄ — C₂³.₃₀D₈

Generators and relations for C₂³.₃₀D₈
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=a, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=ebe^-1=bc=cb, cd=dc, ce=ec, ede^-1=abd^-1 >

Subgroups: 320 in 138 conjugacy classes, 48 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂³×C₄, C₂³×C₄, C₂×C₂.C₄², C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂³.₃₀D₈
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≀C₂, C₄.Q₈, C₂.D₈, C₂².SD₁₆, C₂³.₃₁D₄, C₄²⋊₆C₄, C₂².₄Q₁₆, C₂³.₉D₄, C₂³.₃₀D₈

Smallest permutation representation of C₂³.₃₀D₈
►On 32 points

Generators in S₃₂

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

(2 10)(4 12)(6 14)(8 16)(18 27)(20 29)(22 31)(24 25)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4N	4O	4P	4Q	4R	8A	···	8H
order	1	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4	8	···	8
size	1	1	···	1	2	2	2	2	2	2	2	2	4	···	4	8	8	8	8	4	···	4

38 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4
type	+	+	+	+				+	-	+	+		-		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	D₄	D₈	SD₁₆	Q₁₆	C₄≀C₂	C₂³⋊C₄
kernel	C₂³.₃₀D₈	C₂×C₂.C₄²	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂.C₄²	C₂²⋊C₈	C₂×C₄⋊C₄	C₂²×C₄	C₂²×C₄	C₂⁴	C₂³	C₂³	C₂³	C₂²	C₂²
# reps	1	1	1	1	4	4	4	2	1	1	2	4	2	8	2

Matrix representation of C₂³.₃₀D₈ ►in GL₆(𝔽₁₇)

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

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	0	0	1	0
0	0	0	0	0	16

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	0	0	16	0
0	0	0	0	0	16

3	14	0	0	0	0
3	3	0	0	0	0
0	0	5	12	0	0
0	0	5	5	0	0
0	0	0	0	0	16
0	0	0	0	4	0

3	14	0	0	0	0
14	14	0	0	0	0
0	0	12	5	0	0
0	0	5	5	0	0
0	0	0	0	0	1
0	0	0	0	16	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,4,0,0,0,0,16,0],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,0,16,0,0,0,0,1,0] >;

C₂³.₃₀D₈ in GAP, Magma, Sage, TeX

C_2^3._{30}D_8

% in TeX

G:=Group("C2^3.30D8");

// GroupNames label

G:=SmallGroup(128,26);

// by ID

G=gap.SmallGroup(128,26);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,3924]);

// Polycyclic

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

// generators/relations

G = C23.30D8 order 128 = 27

1st non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₃₀D₈ order 128 = 2⁷

1^st non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂