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G = C6.1462+ 1+4order 192 = 26·3

55th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.1462+ 1+4, (C2×D4)⋊45D6, (C2×C12)⋊16D4, (C2×Q8)⋊37D6, (C22×C4)⋊34D6, C232D632C2, C123D430C2, C127D448C2, D6⋊C438C22, C12.430(C2×D4), (C6×D4)⋊48C22, (C6×Q8)⋊41C22, C2.70(D4○D12), (C22×D12)⋊21C2, (C2×C6).316C24, C4⋊Dic366C22, C6.166(C22×D4), C12.23D432C2, (C2×C12).653C23, (C22×C12)⋊32C22, C37(C22.29C24), (C4×Dic3)⋊45C22, (C2×D12).281C22, C23.26D639C2, (S3×C23).80C22, C23.222(C22×S3), (C22×C6).242C23, C22.325(S3×C23), (C22×S3).138C23, (C2×Dic3).163C23, C6.D4.136C22, (C6×C4○D4)⋊8C2, (C2×C4○D4)⋊12S3, (C2×C4)⋊7(C3⋊D4), (C2×C6).82(C2×D4), C4.33(C2×C3⋊D4), (C2×C3⋊D4)⋊31C22, C2.39(C22×C3⋊D4), C22.24(C2×C3⋊D4), (C2×C4).251(C22×S3), SmallGroup(192,1389)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C6.1462+ 1+4
C1C3C6C2×C6C22×S3S3×C23C22×D12 — C6.1462+ 1+4
C3C2×C6 — C6.1462+ 1+4
C1C22C2×C4○D4

Generators and relations for C6.1462+ 1+4
 G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, dbd-1=a3b, be=eb, cd=dc, ece=a3c, ede=a3b2d >

Subgroups: 1032 in 334 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×22], Q8 [×2], C23, C23 [×2], C23 [×12], Dic3 [×4], C12 [×4], C12 [×2], D6 [×20], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24 [×2], D12 [×8], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×6], C2×C12 [×4], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×4], C22×S3 [×8], C22×C6, C22×C6 [×2], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, C4×Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C2×D12 [×4], C2×D12 [×4], C2×C3⋊D4 [×8], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], S3×C23 [×2], C22.29C24, C23.26D6, C127D4 [×4], C232D6 [×4], C123D4 [×2], C12.23D4 [×2], C22×D12, C6×C4○D4, C6.1462+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×C3⋊D4 [×6], S3×C23, C22.29C24, D4○D12 [×2], C22×C3⋊D4, C6.1462+ 1+4

Smallest permutation representation of C6.1462+ 1+4
On 48 points
Generators in S48
(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 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 12 17 45)(2 7 18 46)(3 8 13 47)(4 9 14 48)(5 10 15 43)(6 11 16 44)(19 39 30 34)(20 40 25 35)(21 41 26 36)(22 42 27 31)(23 37 28 32)(24 38 29 33)
(1 45 4 48)(2 44 5 47)(3 43 6 46)(7 13 10 16)(8 18 11 15)(9 17 12 14)(19 35 22 32)(20 34 23 31)(21 33 24 36)(25 39 28 42)(26 38 29 41)(27 37 30 40)
(1 29 14 21)(2 28 15 20)(3 27 16 19)(4 26 17 24)(5 25 18 23)(6 30 13 22)(7 35 43 37)(8 34 44 42)(9 33 45 41)(10 32 46 40)(11 31 47 39)(12 36 48 38)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)

G:=sub<Sym(48)| (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)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,12,17,45)(2,7,18,46)(3,8,13,47)(4,9,14,48)(5,10,15,43)(6,11,16,44)(19,39,30,34)(20,40,25,35)(21,41,26,36)(22,42,27,31)(23,37,28,32)(24,38,29,33), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,13,10,16)(8,18,11,15)(9,17,12,14)(19,35,22,32)(20,34,23,31)(21,33,24,36)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,29,14,21)(2,28,15,20)(3,27,16,19)(4,26,17,24)(5,25,18,23)(6,30,13,22)(7,35,43,37)(8,34,44,42)(9,33,45,41)(10,32,46,40)(11,31,47,39)(12,36,48,38), (1,21)(2,22)(3,23)(4,24)(5,19)(6,20)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)>;

G:=Group( (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)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,12,17,45)(2,7,18,46)(3,8,13,47)(4,9,14,48)(5,10,15,43)(6,11,16,44)(19,39,30,34)(20,40,25,35)(21,41,26,36)(22,42,27,31)(23,37,28,32)(24,38,29,33), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,13,10,16)(8,18,11,15)(9,17,12,14)(19,35,22,32)(20,34,23,31)(21,33,24,36)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,29,14,21)(2,28,15,20)(3,27,16,19)(4,26,17,24)(5,25,18,23)(6,30,13,22)(7,35,43,37)(8,34,44,42)(9,33,45,41)(10,32,46,40)(11,31,47,39)(12,36,48,38), (1,21)(2,22)(3,23)(4,24)(5,19)(6,20)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45) );

G=PermutationGroup([(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),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,12,17,45),(2,7,18,46),(3,8,13,47),(4,9,14,48),(5,10,15,43),(6,11,16,44),(19,39,30,34),(20,40,25,35),(21,41,26,36),(22,42,27,31),(23,37,28,32),(24,38,29,33)], [(1,45,4,48),(2,44,5,47),(3,43,6,46),(7,13,10,16),(8,18,11,15),(9,17,12,14),(19,35,22,32),(20,34,23,31),(21,33,24,36),(25,39,28,42),(26,38,29,41),(27,37,30,40)], [(1,29,14,21),(2,28,15,20),(3,27,16,19),(4,26,17,24),(5,25,18,23),(6,30,13,22),(7,35,43,37),(8,34,44,42),(9,33,45,41),(10,32,46,40),(11,31,47,39),(12,36,48,38)], [(1,21),(2,22),(3,23),(4,24),(5,19),(6,20),(7,42),(8,37),(9,38),(10,39),(11,40),(12,41),(13,28),(14,29),(15,30),(16,25),(17,26),(18,27),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)])

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D···6I12A12B12C12D12E···12J
order122222222222344444444446666···61212121212···12
size11112244121212122222244121212122224···422224···4

42 irreducible representations

dim1111111122222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D6D6D6C3⋊D42+ 1+4D4○D12
kernelC6.1462+ 1+4C23.26D6C127D4C232D6C123D4C12.23D4C22×D12C6×C4○D4C2×C4○D4C2×C12C22×C4C2×D4C2×Q8C2×C4C6C2
# reps1144221114331824

Matrix representation of C6.1462+ 1+4 in GL6(𝔽13)

110000
1200000
0001200
0011200
0000012
0000112
,
240000
9110000
0010661
0073127
000037
0000610
,
240000
2110000
0061016
0037712
000073
0000106
,
240000
2110000
0001202
0012020
0001201
0012010
,
240000
9110000
0010110
0001011
0000120
0000012

G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,6,12,3,6,0,0,1,7,7,10],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,6,3,0,0,0,0,10,7,0,0,0,0,1,7,7,10,0,0,6,12,3,6],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,0,12,0,12,0,0,12,0,12,0,0,0,0,2,0,1,0,0,2,0,1,0],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,0,12,0,0,0,0,11,0,12] >;

C6.1462+ 1+4 in GAP, Magma, Sage, TeX

C_6._{146}2_+^{1+4}
% in TeX

G:=Group("C6.146ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1389);
// by ID

G=gap.SmallGroup(192,1389);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=a^3*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,d*b*d^-1=a^3*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
// generators/relations

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