Copied to
clipboard

## G = C2×C22.19C24order 128 = 27

### Direct product of C2 and C22.19C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.19C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C24×C4 — C2×C22.19C24
 Lower central C1 — C22 — C2×C22.19C24
 Upper central C1 — C22×C4 — C2×C22.19C24
 Jennings C1 — C22 — C2×C22.19C24

Generators and relations for C2×C22.19C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1452 in 940 conjugacy classes, 452 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C25, C2×C42⋊C2, C2×C4×D4, C2×C22≀C2, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4, C22.19C24, C24×C4, C22×C4○D4, C2×C22.19C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C25, C22.19C24, D4×C23, C22×C4○D4, C2×C22.19C24

Smallest permutation representation of C2×C22.19C24
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 19)(14 20)(15 17)(16 18)(25 30)(26 31)(27 32)(28 29)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 23)(2 24)(3 21)(4 22)(5 10)(6 11)(7 12)(8 9)(13 27)(14 28)(15 25)(16 26)(17 30)(18 31)(19 32)(20 29)
(1 18)(2 19)(3 20)(4 17)(5 27)(6 28)(7 25)(8 26)(9 16)(10 13)(11 14)(12 15)(21 29)(22 30)(23 31)(24 32)
(1 23)(2 24)(3 21)(4 22)(5 10)(6 11)(7 12)(8 9)(13 25)(14 26)(15 27)(16 28)(17 32)(18 29)(19 30)(20 31)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 25)(14 26)(15 27)(16 28)(17 32)(18 29)(19 30)(20 31)(21 23)(22 24)
(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)| (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,19)(14,20)(15,17)(16,18)(25,30)(26,31)(27,32)(28,29), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,27)(14,28)(15,25)(16,26)(17,30)(18,31)(19,32)(20,29), (1,18)(2,19)(3,20)(4,17)(5,27)(6,28)(7,25)(8,26)(9,16)(10,13)(11,14)(12,15)(21,29)(22,30)(23,31)(24,32), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,25)(14,26)(15,27)(16,28)(17,32)(18,29)(19,30)(20,31), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,25)(14,26)(15,27)(16,28)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24), (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( (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,19)(14,20)(15,17)(16,18)(25,30)(26,31)(27,32)(28,29), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,27)(14,28)(15,25)(16,26)(17,30)(18,31)(19,32)(20,29), (1,18)(2,19)(3,20)(4,17)(5,27)(6,28)(7,25)(8,26)(9,16)(10,13)(11,14)(12,15)(21,29)(22,30)(23,31)(24,32), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,25)(14,26)(15,27)(16,28)(17,32)(18,29)(19,30)(20,31), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,25)(14,26)(15,27)(16,28)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24), (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([[(1,9),(2,10),(3,11),(4,12),(5,24),(6,21),(7,22),(8,23),(13,19),(14,20),(15,17),(16,18),(25,30),(26,31),(27,32),(28,29)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,23),(2,24),(3,21),(4,22),(5,10),(6,11),(7,12),(8,9),(13,27),(14,28),(15,25),(16,26),(17,30),(18,31),(19,32),(20,29)], [(1,18),(2,19),(3,20),(4,17),(5,27),(6,28),(7,25),(8,26),(9,16),(10,13),(11,14),(12,15),(21,29),(22,30),(23,31),(24,32)], [(1,23),(2,24),(3,21),(4,22),(5,10),(6,11),(7,12),(8,9),(13,25),(14,26),(15,27),(16,28),(17,32),(18,29),(19,30),(20,31)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,25),(14,26),(15,27),(16,28),(17,32),(18,29),(19,30),(20,31),(21,23),(22,24)], [(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)]])

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T 2U 2V 2W 4A ··· 4H 4I ··· 4T 4U ··· 4AF order 1 2 ··· 2 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 kernel C2×C22.19C24 C2×C42⋊C2 C2×C4×D4 C2×C22≀C2 C2×C4⋊D4 C2×C22⋊Q8 C2×C22.D4 C22.19C24 C24×C4 C22×C4○D4 C22×C4 C23 # reps 1 1 4 2 2 2 2 16 1 1 8 16

Matrix representation of C2×C22.19C24 in GL6(𝔽5)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 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 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 1 2 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 1 1
,
 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 1 1
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 3 0 0 0 0 0 0 3

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,2,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C2×C22.19C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{19}C_2^4
% in TeX

G:=Group("C2xC2^2.19C2^4");
// GroupNames label

G:=SmallGroup(128,2167);
// by ID

G=gap.SmallGroup(128,2167);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,136]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations

׿
×
𝔽