C2^4.7Q8 - GroupNames

G = C₂⁴.₇Q₈ order 128 = 2⁷

6^th non-split extension by C₂⁴ of Q₈ acting via Q₈/C₂=C₂²

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

Aliases: C₂⁴.₇Q₈, (C₂×C₄).₂₁C₄², C₄.₂₀(C₂×C₄²), (C₂²×C₄).₃₇Q₈, C₂³.₇₆(C₂×Q₈), C₂³.₂₃(C₄⋊C₄), (C₂²×C₄).₂₅₆D₄, C₄.C₄²⋊₁₆C₂, M₄(2).₂₈(C₂×C₄), (C₂×M₄(2)).₂₆C₄, (C₂³×C₄).₂₁₉C₂², (C₂²×C₈).₃₇₃C₂², C₂.₂(M₄(2).C₄), C₄.₂₀(C₂.C₄²), (C₂²×C₄).₁₃₀₄C₂³, (C₂²×M₄(2)).₁₄C₂, (C₂×M₄(2)).₂₉₈C₂², C₂².₁₂(C₂.C₄²), (C₂×C₄).₄₀(C₄⋊C₄), (C₂×C₈).₁₂₇(C₂×C₄), C₂².₁₁(C₂×C₄⋊C₄), C₄.₈₃(C₂×C₂²⋊C₄), (C₂×C₄).₁₂₉₅(C₂×D₄), (C₂×C₄).₅₁₈(C₂²×C₄), (C₂²×C₄).₂₅₅(C₂×C₄), (C₂×C₄).₂₅₃(C₂²⋊C₄), C₂.₁₅(C₂×C₂.C₄²), SmallGroup(128,470)

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

Derived series

C₁ — C₄ — C₂⁴.₇Q₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×M₄(2) — C₂⁴.₇Q₈

Lower central

C₁ — C₂ — C₄ — C₂⁴.₇Q₈

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂⁴.₇Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₇Q₈

Generators and relations for C₂⁴.₇Q₈
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=cde², ab=ba, ac=ca, eae^-1=ad=da, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=bde³ >

Subgroups: 276 in 186 conjugacy classes, 108 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂⁴, C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₄.C₄², C₂²×M₄(2), C₂²×M₄(2), C₂⁴.₇Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₂.C₄², M₄(2).C₄, C₂⁴.₇Q₈

Smallest permutation representation of C₂⁴.₇Q₈
►On 32 points

Generators in S₃₂

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

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

(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4J	8A	···	8X
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	8	···	8
size	1	1	1	1	2	···	2	1	1	1	1	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	2	2	2	4
type	+	+	+		+	-	-
image	C₁	C₂	C₂	C₄	D₄	Q₈	Q₈	M₄(2).C₄
kernel	C₂⁴.₇Q₈	C₄.C₄²	C₂²×M₄(2)	C₂×M₄(2)	C₂²×C₄	C₂²×C₄	C₂⁴	C₂
# reps	1	4	3	24	6	1	1	4

Matrix representation of C₂⁴.₇Q₈ ►in GL₆(𝔽₁₇)

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

16	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	0
0	0	0	0	0	16

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	4	0	16	0
0	0	1	0	0	16

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	13	15	0	0
0	0	6	4	0	0
0	0	7	13	0	4
0	0	0	16	16	0

5	3	0	0	0	0
14	12	0	0	0	0
0	0	4	0	15	0
0	0	0	0	4	1
0	0	10	0	13	0
0	0	11	13	16	0

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

C₂⁴.₇Q₈ in GAP, Magma, Sage, TeX

C_2^4._7Q_8

% in TeX

G:=Group("C2^4.7Q8");

// GroupNames label

G:=SmallGroup(128,470);

// by ID

G=gap.SmallGroup(128,470);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,2019,248,172,4037,124]);

// Polycyclic

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

// generators/relations

G = C24.7Q8 order 128 = 27

6th non-split extension by C24 of Q8 acting via Q8/C2=C22

G = C₂⁴.₇Q₈ order 128 = 2⁷

6^th non-split extension by C₂⁴ of Q₈ acting via Q₈/C₂=C₂²