C2^4.73(C2xC4) - GroupNames

G = C₂⁴.₇₃(C₂×C₄) order 128 = 2⁷

38^th non-split extension by C₂⁴ of C₂×C₄ acting via C₂×C₄/C₂=C₂²

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴.₇₃(C₂×C₄), C₂²⋊C₈⋊₄₃C₂², (C₂×C₄).₆₂₉C₂⁴, (C₂×C₈).₃₈₉C₂³, (C₂²×C₈)⋊₄₅C₂², (C₂²×D₄).₃₄C₄, C₄.₁₇₈(C₂²×D₄), (C₂²×C₄).₄₁₃D₄, (C₂²×Q₈).₂₇C₄, C₂⁴.₄C₄⋊₃₀C₂, C₂.₆(Q₈○M₄(2)), C₂³.₉₆(C₂²×C₄), C₂³.₈₆(C₂²⋊C₄), (C₂×M₄(2))⋊₆₄C₂², (C₂²×M₄(2))⋊₁₅C₂, C₂².₁₅₉(C₂³×C₄), (C₂³×C₄).₅₀₆C₂², (C₂²×C₄).₁₄₉₃C₂³, (C₂×C₄○D₄).₂₂C₄, (C₂×C₄).₈₃₉(C₂×D₄), C₄.₆₇(C₂×C₂²⋊C₄), (C₂×D₄).₂₁₉(C₂×C₄), (C₂×Q₈).₁₉₈(C₂×C₄), (C₂²×C₈)⋊C₂⋊₂₆C₂, (C₂²×C₄).₃₁₉(C₂×C₄), (C₂×C₄).₂₄₀(C₂²×C₄), (C₂²×C₄○D₄).₁₄C₂, C₂².₁₈(C₂×C₂²⋊C₄), C₂.₂₂(C₂²×C₂²⋊C₄), (C₂×C₄).₁₅₅(C₂²⋊C₄), (C₂×C₄○D₄).₂₆₈C₂², SmallGroup(128,1611)

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

Derived series

C₁ — C₂² — C₂⁴.₇₃(C₂×C₄)

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂⁴.₇₃(C₂×C₄)

Lower central

C₁ — C₂² — C₂⁴.₇₃(C₂×C₄)

Upper central

C₁ — C₂×C₄ — C₂⁴.₇₃(C₂×C₄)

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₇₃(C₂×C₄)

Generators and relations for C₂⁴.₇₃(C₂×C₄)
G = < a,b,c,d,e,f | a²=b²=c²=d²=e²=1, f⁴=d, ab=ba, faf^-1=ac=ca, eae=ad=da, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 636 in 382 conjugacy classes, 172 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂²⋊C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂⁴.₄C₄, (C₂²×C₈)⋊C₂, C₂²×M₄(2), C₂²×C₄○D₄, C₂⁴.₇₃(C₂×C₄)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×C₂²⋊C₄, Q₈○M₄(2), C₂⁴.₇₃(C₂×C₄)

Smallest permutation representation of C₂⁴.₇₃(C₂×C₄)
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+				+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈○M₄(2)
kernel	C₂⁴.₇₃(C₂×C₄)	C₂⁴.₄C₄	(C₂²×C₈)⋊C₂	C₂²×M₄(2)	C₂²×C₄○D₄	C₂²×D₄	C₂²×Q₈	C₂×C₄○D₄	C₂²×C₄	C₂
# reps	1	4	8	2	1	6	2	8	8	4

Matrix representation of C₂⁴.₇₃(C₂×C₄) ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	1	0	0	0	0
0	0	0	4	0	0
0	0	13	0	0	0
0	0	0	0	0	4
0	0	0	0	13	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	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	0	0	1	0
0	0	0	0	0	1

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	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

0	16	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	4	0	0	0
0	0	0	4	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂⁴.₇₃(C₂×C₄) in GAP, Magma, Sage, TeX

C_2^4._{73}(C_2\times C_4)

% in TeX

G:=Group("C2^4.73(C2xC4)");

// GroupNames label

G:=SmallGroup(128,1611);

// by ID

G=gap.SmallGroup(128,1611);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,2019,124]);

// Polycyclic

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

// generators/relations

G = C24.73(C2×C4) order 128 = 27

38th non-split extension by C24 of C2×C4 acting via C2×C4/C2=C22

G = C₂⁴.₇₃(C₂×C₄) order 128 = 2⁷

38^th non-split extension by C₂⁴ of C₂×C₄ acting via C₂×C₄/C₂=C₂²