C4^2.5Q8 - GroupNames

G = C₄².₅Q₈ order 128 = 2⁷

5^th non-split extension by C₄² of Q₈ acting via Q₈/C₂=C₂²

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

Aliases: C₄².₅Q₈, C₄².₂₂D₄, C₄⋊C₈⋊₂C₄, (C₄×C₈)⋊₂C₄, (C₂×C₄).₈D₈, (C₂×C₄).₄Q₁₆, C₄.₇(C₄.Q₈), C₄.₈(C₂.D₈), (C₂×C₄).₃₀C₄², (C₂×C₄).₁₁SD₁₆, C₄²⋊₈C₄.₁C₂, C₄².₃₀₄(C₂×C₄), (C₂²×C₄).₁₇₆D₄, C₄.₂₈(D₄⋊C₄), C₂.₅(C₄.₉C₄²), C₄.₂₀(Q₈⋊C₄), C₄⋊M₄(2).₄C₂, C₂.C₄².₁C₄, C₄².₁₂C₄.₇C₂, C₂.₃(C₂².₄Q₁₆), (C₂×C₄²).₁₂₆C₂², C₂².₁₀(D₄⋊C₄), C₂.₄(M₄(2)⋊₄C₄), C₂³.₁₃₈(C₂²⋊C₄), C₂².₁₃(Q₈⋊C₄), C₂².₃₆(C₂.C₄²), (C₂×C₄).₉₅(C₄⋊C₄), (C₂²×C₄).₁₄₆(C₂×C₄), (C₂×C₄).₃₃₆(C₂²⋊C₄), SmallGroup(128,18)

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

Derived series

C₁ — C₂×C₄ — C₄².₅Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄².₁₂C₄ — C₄².₅Q₈

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₅Q₈

Upper central

C₁ — C₂² — C₂×C₄² — C₄².₅Q₈

Jennings

C₁ — C₂² — C₂² — C₂×C₄² — C₄².₅Q₈

Generators and relations for C₄².₅Q₈
G = < a,b,c,d | a⁴=b⁴=1, c⁴=b², d²=b^-1c², ab=ba, ac=ca, dad^-1=a^-1, cbc^-1=a²b, dbd^-1=a²b^-1, dcd^-1=a^-1b²c³ >

Subgroups: 160 in 78 conjugacy classes, 40 normal (34 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₄², C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂.C₄², C₂.C₄², C₄×C₈, C₂²⋊C₈, C₄⋊C₈, C₄⋊C₈, C₂×C₄², C₂×C₄⋊C₄, C₂×M₄(2), C₄²⋊₈C₄, C₄⋊M₄(2), C₄².₁₂C₄, C₄².₅Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, D₈, SD₁₆, Q₁₆, C₂.C₄², D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄.₉C₄², C₂².₄Q₁₆, M₄(2)⋊₄C₄, C₄².₅Q₈

Smallest permutation representation of C₄².₅Q₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+				+	-	+	+		-
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	D₄	D₈	SD₁₆	Q₁₆	C₄.₉C₄²	M₄(2)⋊₄C₄
kernel	C₄².₅Q₈	C₄²⋊₈C₄	C₄⋊M₄(2)	C₄².₁₂C₄	C₂.C₄²	C₄×C₈	C₄⋊C₈	C₄²	C₄²	C₂²×C₄	C₂×C₄	C₂×C₄	C₂×C₄	C₂	C₂
# reps	1	1	1	1	4	4	4	1	1	2	2	4	2	2	2

Matrix representation of C₄².₅Q₈ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	9	0	0
0	0	13	1	0	0
0	0	0	0	16	9
0	0	0	0	13	1

4	15	0	0	0	0
0	13	0	0	0	0
0	0	13	2	0	0
0	0	1	4	0	0
0	0	3	12	4	15
0	0	6	0	16	13

8	7	0	0	0	0
0	2	0	0	0	0
0	0	0	3	15	0
0	0	10	12	0	15
0	0	0	0	0	14
0	0	0	0	7	5

13	0	0	0	0	0
1	4	0	0	0	0
0	0	1	0	0	0
0	0	4	16	0	0
0	0	6	1	4	0
0	0	15	11	16	13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,13,0,0,0,0,9,1,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[4,0,0,0,0,0,15,13,0,0,0,0,0,0,13,1,3,6,0,0,2,4,12,0,0,0,0,0,4,16,0,0,0,0,15,13],[8,0,0,0,0,0,7,2,0,0,0,0,0,0,0,10,0,0,0,0,3,12,0,0,0,0,15,0,0,7,0,0,0,15,14,5],[13,1,0,0,0,0,0,4,0,0,0,0,0,0,1,4,6,15,0,0,0,16,1,11,0,0,0,0,4,16,0,0,0,0,0,13] >;

C₄².₅Q₈ in GAP, Magma, Sage, TeX

C_4^2._5Q_8

% in TeX

G:=Group("C4^2.5Q8");

// GroupNames label

G:=SmallGroup(128,18);

// by ID

G=gap.SmallGroup(128,18);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924,102]);

// Polycyclic

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

// generators/relations

G = C42.5Q8 order 128 = 27

5th non-split extension by C42 of Q8 acting via Q8/C2=C22

G = C₄².₅Q₈ order 128 = 2⁷

5^th non-split extension by C₄² of Q₈ acting via Q₈/C₂=C₂²