C4^2.6Q8 - GroupNames

G = C₄².₆Q₈ order 128 = 2⁷

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

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

Aliases: C₄².₆Q₈, C₄².₂₄D₄, (C₄×C₈)⋊₇C₄, C₈⋊C₄⋊₄C₄, C₂².₁₀C₄≀C₂, (C₂×C₄).₃₂C₄², C₄²⋊₅C₄.₁C₂, C₄².₂₉₀(C₂×C₄), (C₂²×C₄).₁₇₇D₄, (C₄×M₄(2)).₉C₂, C₂.₇(C₄.₉C₄²), C₂.₆(C₄²⋊₆C₄), C₂.C₄².₂C₄, C₄².₆C₄.₆C₂, (C₂×C₄²).₁₂₈C₂², C₂.₆(M₄(2)⋊₄C₄), C₂³.₁₃₉(C₂²⋊C₄), C₂².₃₈(C₂.C₄²), (C₂×C₄).₁₅(C₄⋊C₄), (C₂²×C₄).₁₄₇(C₂×C₄), (C₂×C₄).₂₉₅(C₂²⋊C₄), SmallGroup(128,20)

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

Derived series

C₁ — C₂×C₄ — C₄².₆Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×M₄(2) — C₄².₆Q₈

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₆Q₈

Upper central

C₁ — C₂² — C₂×C₄² — C₄².₆Q₈

Jennings

C₁ — C₂² — C₂² — C₂×C₄² — C₄².₆Q₈

Generators and relations for C₄².₆Q₈
G = < a,b,c,d | a⁴=b⁴=1, c⁴=b², d²=b^-1c², ab=ba, ac=ca, dad^-1=ab², cbc^-1=b^-1, dbd^-1=a²b^-1, dcd^-1=ac³ >

Subgroups: 152 in 76 conjugacy classes, 32 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₄², C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂.C₄², C₂.C₄², C₄×C₈, C₈⋊C₄, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₂×M₄(2), C₄²⋊₅C₄, C₄×M₄(2), C₄².₆C₄, C₄².₆Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂.C₄², C₄≀C₂, C₄.₉C₄², C₄²⋊₆C₄, M₄(2)⋊₄C₄, C₄².₆Q₈

Smallest permutation representation of C₄².₆Q₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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	4	4
type	+	+	+	+				+	-	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	D₄	C₄≀C₂	C₄.₉C₄²	M₄(2)⋊₄C₄
kernel	C₄².₆Q₈	C₄²⋊₅C₄	C₄×M₄(2)	C₄².₆C₄	C₂.C₄²	C₄×C₈	C₈⋊C₄	C₄²	C₄²	C₂²×C₄	C₂²	C₂	C₂
# reps	1	1	1	1	4	4	4	1	1	2	8	2	2

Matrix representation of C₄².₆Q₈ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	13	0	0	0	0
0	0	1	9	0	0
0	0	0	16	0	0
0	0	0	3	1	4
0	0	0	0	0	16

0	1	0	0	0	0
1	0	0	0	0	0
0	0	4	2	0	0
0	0	0	13	0	0
0	0	14	12	13	1
0	0	0	0	0	4

10	11	0	0	0	0
11	10	0	0	0	0
0	0	12	0	15	0
0	0	0	0	0	1
0	0	2	16	5	0
0	0	0	13	0	0

4	0	0	0	0	0
0	13	0	0	0	0
0	0	1	0	0	0
0	0	13	16	0	0
0	0	16	0	4	0
0	0	12	0	15	13

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

C₄².₆Q₈ in GAP, Magma, Sage, TeX

C_4^2._6Q_8

% in TeX

G:=Group("C4^2.6Q8");

// GroupNames label

G:=SmallGroup(128,20);

// by ID

G=gap.SmallGroup(128,20);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924,172]);

// 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*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*c^3>;

// generators/relations

G = C42.6Q8 order 128 = 27

6th non-split extension by C42 of Q8 acting via Q8/C2=C22

G = C₄².₆Q₈ order 128 = 2⁷

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