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G = 2- 1+43C6order 192 = 26·3

2nd semidirect product of 2- 1+4 and C6 acting via C6/C2=C3

non-abelian, soluble

Aliases: 2- 1+43C6, 2+ 1+44C6, SL2(𝔽3).11C23, C4○D4.A4, D4.A46C2, D4.5(C2×A4), Q8.A46C2, Q8.7(C2×A4), C4.A48C22, C2.9(C23×A4), C4.11(C22×A4), C2.C251C3, Q8.4(C22×C6), C22.9(C22×A4), (C2×SL2(𝔽3))⋊3C22, (C2×Q8).(C2×C6), (C2×C4.A4)⋊9C2, (C2×C4○D4)⋊3C6, (C2×C4).14(C2×A4), C4○D4.5(C2×C6), SmallGroup(192,1504)

Series: Derived Chief Lower central Upper central

C1C2Q8 — 2- 1+43C6
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3)C2×C4.A4 — 2- 1+43C6
Q8 — 2- 1+43C6
C1C4C4○D4

Generators and relations for 2- 1+43C6
 G = < a,b,c,d,e | a4=b2=e6=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a2b, dcd-1=a2c, ece-1=cd, ede-1=c >

Subgroups: 573 in 195 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×9], C6 [×4], C2×C4 [×3], C2×C4 [×19], D4 [×3], D4 [×19], Q8 [×2], Q8 [×6], C23 [×5], C12 [×4], C2×C6 [×3], C22×C4 [×5], C2×D4 [×15], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×2], C4○D4 [×3], C4○D4 [×25], SL2(𝔽3), C2×C12 [×3], C3×D4 [×3], C3×Q8, C2×C4○D4 [×3], C2×C4○D4 [×4], 2+ 1+4, 2+ 1+4 [×3], 2- 1+4 [×3], 2- 1+4, C2×SL2(𝔽3) [×3], C4.A4, C4.A4 [×3], C3×C4○D4, C2.C25, C2×C4.A4 [×3], Q8.A4, D4.A4 [×3], 2- 1+43C6
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, A4, C2×C6 [×7], C2×A4 [×7], C22×C6, C22×A4 [×7], C23×A4, 2- 1+43C6

Smallest permutation representation of 2- 1+43C6
On 32 points
Generators in S32
(1 3 2 4)(5 7 6 8)(9 13 19 15)(10 14 20 16)(11 12 18 17)(21 29 24 32)(22 30 25 27)(23 31 26 28)
(1 7)(2 8)(3 5)(4 6)(9 23)(10 21)(11 25)(12 30)(13 28)(14 32)(15 31)(16 29)(17 27)(18 22)(19 26)(20 24)
(1 12 2 17)(3 18 4 11)(5 22 6 25)(7 30 8 27)(9 20 19 10)(13 16 15 14)(21 23 24 26)(28 29 31 32)
(1 13 2 15)(3 19 4 9)(5 26 6 23)(7 28 8 31)(10 18 20 11)(12 14 17 16)(21 22 24 25)(27 29 30 32)
(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)| (1,3,2,4)(5,7,6,8)(9,13,19,15)(10,14,20,16)(11,12,18,17)(21,29,24,32)(22,30,25,27)(23,31,26,28), (1,7)(2,8)(3,5)(4,6)(9,23)(10,21)(11,25)(12,30)(13,28)(14,32)(15,31)(16,29)(17,27)(18,22)(19,26)(20,24), (1,12,2,17)(3,18,4,11)(5,22,6,25)(7,30,8,27)(9,20,19,10)(13,16,15,14)(21,23,24,26)(28,29,31,32), (1,13,2,15)(3,19,4,9)(5,26,6,23)(7,28,8,31)(10,18,20,11)(12,14,17,16)(21,22,24,25)(27,29,30,32), (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( (1,3,2,4)(5,7,6,8)(9,13,19,15)(10,14,20,16)(11,12,18,17)(21,29,24,32)(22,30,25,27)(23,31,26,28), (1,7)(2,8)(3,5)(4,6)(9,23)(10,21)(11,25)(12,30)(13,28)(14,32)(15,31)(16,29)(17,27)(18,22)(19,26)(20,24), (1,12,2,17)(3,18,4,11)(5,22,6,25)(7,30,8,27)(9,20,19,10)(13,16,15,14)(21,23,24,26)(28,29,31,32), (1,13,2,15)(3,19,4,9)(5,26,6,23)(7,28,8,31)(10,18,20,11)(12,14,17,16)(21,22,24,25)(27,29,30,32), (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([(1,3,2,4),(5,7,6,8),(9,13,19,15),(10,14,20,16),(11,12,18,17),(21,29,24,32),(22,30,25,27),(23,31,26,28)], [(1,7),(2,8),(3,5),(4,6),(9,23),(10,21),(11,25),(12,30),(13,28),(14,32),(15,31),(16,29),(17,27),(18,22),(19,26),(20,24)], [(1,12,2,17),(3,18,4,11),(5,22,6,25),(7,30,8,27),(9,20,19,10),(13,16,15,14),(21,23,24,26),(28,29,31,32)], [(1,13,2,15),(3,19,4,9),(5,26,6,23),(7,28,8,31),(10,18,20,11),(12,14,17,16),(21,22,24,25),(27,29,30,32)], [(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)])

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H3A3B4A4B4C4D4E4F4G4H4I6A6B6C···6H12A12B12C12D12E···12J
order12222222233444444444666···61212121212···12
size11222666644112226666448···844448···8

38 irreducible representations

dim1111111133334
type++++++++
imageC1C2C2C2C3C6C6C6A4C2×A4C2×A4C2×A42- 1+43C6
kernel2- 1+43C6C2×C4.A4Q8.A4D4.A4C2.C25C2×C4○D42+ 1+42- 1+4C4○D4C2×C4D4Q8C1
# reps1313262613316

Matrix representation of 2- 1+43C6 in GL4(𝔽5) generated by

0300
3000
0203
3030
,
0200
3000
0302
3030
,
2020
0002
0030
0200
,
2000
0200
1030
0003
,
0202
4020
0402
0040
G:=sub<GL(4,GF(5))| [0,3,0,3,3,0,2,0,0,0,0,3,0,0,3,0],[0,3,0,3,2,0,3,0,0,0,0,3,0,0,2,0],[2,0,0,0,0,0,0,2,2,0,3,0,0,2,0,0],[2,0,1,0,0,2,0,0,0,0,3,0,0,0,0,3],[0,4,0,0,2,0,4,0,0,2,0,4,2,0,2,0] >;

2- 1+43C6 in GAP, Magma, Sage, TeX

2_-^{1+4}\rtimes_3C_6
% in TeX

G:=Group("ES-(2,2):3C6");
// GroupNames label

G:=SmallGroup(192,1504);
// by ID

G=gap.SmallGroup(192,1504);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,235,172,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=e^6=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*b,d*c*d^-1=a^2*c,e*c*e^-1=c*d,e*d*e^-1=c>;
// generators/relations

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