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G = C2×C22.19C24order 128 = 27

Direct product of C2 and C22.19C24

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

Aliases: C2×C22.19C24, C429C23, C23.12C24, C22.24C25, C25.96C22, C24.608C23, (C24×C4)⋊9C2, C4⋊C416C23, (C22×C4)⋊56D4, C2.8(D4×C23), (C4×D4)⋊89C22, (C2×D4)⋊15C23, C237(C4○D4), (C2×C4).29C24, (C2×Q8)⋊14C23, C4⋊D495C22, C22⋊C417C23, (C2×C42)⋊42C22, (C22×C4)⋊12C23, (C23×C4)⋊28C22, C23.707(C2×D4), C4.169(C22×D4), C22≀C241C22, C4(C22.19C24), (C22×D4)⋊58C22, C22⋊Q8103C22, (C22×Q8)⋊56C22, C22.46(C22×D4), C42⋊C283C22, C22.D466C22, (C2×C4×D4)⋊69C2, C4(C2×C4⋊D4), C4(C2×C22⋊Q8), C4(C2×C22≀C2), (C2×C4)⋊22(C2×D4), (C2×C4)2C22≀C2, C221(C2×C4○D4), (C2×C4)2(C4⋊D4), (C2×C4⋊D4)⋊76C2, (C22×C4)C22≀C2, (C2×C4)2(C22⋊Q8), (C2×C22⋊Q8)⋊86C2, (C2×C22≀C2)⋊30C2, C2.8(C22×C4○D4), C4(C2×C22.D4), (C2×C4⋊C4)⋊125C22, (C22×C4○D4)⋊11C2, (C2×C4○D4)⋊65C22, (C2×C42⋊C2)⋊51C2, (C22×C4)(C22⋊Q8), (C2×C22⋊C4)⋊83C22, (C2×C4)(C22.19C24), (C2×C4)2(C22.D4), (C2×C22.D4)⋊67C2, (C2×C4)2(C2×C22⋊Q8), (C2×C4)2(C2×C22≀C2), (C22×C4)(C2×C22⋊Q8), (C22×C4)(C2×C22≀C2), SmallGroup(128,2167)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C22.19C24
C1C2C22C23C22×C4C23×C4C24×C4 — C2×C22.19C24
C1C22 — C2×C22.19C24
C1C22×C4 — C2×C22.19C24
C1C22 — 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 [×6], C2 [×16], C4 [×8], C4 [×16], C22, C22 [×18], C22 [×88], C2×C4 [×44], C2×C4 [×72], D4 [×56], Q8 [×8], C23, C23 [×22], C23 [×80], C42 [×8], C22⋊C4 [×40], C4⋊C4 [×24], C22×C4 [×2], C22×C4 [×48], C22×C4 [×60], C2×D4 [×28], C2×D4 [×28], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×8], C24 [×12], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×6], C42⋊C2 [×8], C4×D4 [×32], C22≀C2 [×16], C4⋊D4 [×16], C22⋊Q8 [×16], C22.D4 [×16], C23×C4 [×2], C23×C4 [×14], C23×C4 [×8], C22×D4, C22×D4 [×6], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C25, C2×C42⋊C2, C2×C4×D4 [×4], C2×C22≀C2 [×2], C2×C4⋊D4 [×2], C2×C22⋊Q8 [×2], C2×C22.D4 [×2], C22.19C24 [×16], C24×C4, C22×C4○D4, C2×C22.19C24
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×8], C24 [×31], C22×D4 [×14], C2×C4○D4 [×12], C25, C22.19C24 [×4], D4×C23, C22×C4○D4 [×2], 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···2G2H···2S2T2U2V2W4A···4H4I···4T4U···4AF
order12···22···222224···44···44···4
size11···12···244441···12···24···4

56 irreducible representations

dim111111111122
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4
kernelC2×C22.19C24C2×C42⋊C2C2×C4×D4C2×C22≀C2C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C22.19C24C24×C4C22×C4○D4C22×C4C23
# reps11422221611816

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

400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
000400
004000
000012
000004
,
400000
040000
004000
000400
000040
000011
,
100000
040000
001000
000400
000040
000011
,
400000
040000
004000
000400
000030
000003

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

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