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G = C6.382+ 1+4  order 192 = 26·3

38th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.382+ 1+4, C4⋊C4⋊22D6, (C4×S3)⋊1D4, (C2×D4)⋊7D6, C23⋊2D6⋊9C2, C4⋊D4⋊10S3, C22⋊C4⋊10D6, D6.41(C2×D4), C4.183(S3×D4), (C22×C4)⋊19D6, D6⋊3D4⋊18C2, D6⋊D4⋊12C2, C12⋊D4⋊21C2, C12⋊3D4⋊16C2, C12⋊7D4⋊33C2, D6⋊C4⋊17C22, C12.227(C2×D4), (C6×D4)⋊12C22, Dic3.6(C2×D4), C6.66(C22×D4), C2.27(D4○D12), (C2×D12)⋊23C22, (C2×C6).151C24, C4⋊Dic3⋊31C22, C2.40(D4⋊6D6), (C2×C12).173C23, (C22×C12)⋊20C22, C3⋊3(C22.29C24), (C2×Dic6)⋊54C22, (C4×Dic3)⋊21C22, (C22×C6).20C23, C23.11D6⋊18C2, C6.D4⋊23C22, (S3×C23).46C22, C23.122(C22×S3), C22.172(S3×C23), (C2×Dic3).72C23, (C22×S3).186C23, (C2×S3×D4)⋊10C2, C2.39(C2×S3×D4), (S3×C2×C4)⋊13C22, C4⋊C4⋊7S3⋊20C2, (C2×C4○D12)⋊20C2, (C3×C4⋊D4)⋊13C2, (C3×C4⋊C4)⋊10C22, (C2×C3⋊D4)⋊14C22, (C2×C4).38(C22×S3), (C3×C22⋊C4)⋊12C22, SmallGroup(192,1166)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C6.382+ 1+4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C6.382+ 1+4
Lower central
C3 — C2×C6 — C6.382+ 1+4
Upper central
C1 — C22 — C4⋊D4

Generators and relations for C6.382+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=a3c, ede=b2d >

Subgroups: 1072 in 334 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C42⋊C2, C22≀C2, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C4×Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, C4○D12, S3×D4, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, C22.29C24, D6⋊D4, C23.11D6, C4⋊C4⋊7S3, C12⋊D4, C12⋊7D4, C23⋊2D6, D6⋊3D4, C12⋊3D4, C3×C4⋊D4, C2×C4○D12, C2×S3×D4, C6.382+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, C22.29C24, C2×S3×D4, D4⋊6D6, D4○D12, C6.382+ 1+4

Smallest permutation representation of C6.382+ 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 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 37 22 43)(17 38 23 44)(18 39 24 45)

(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E6F6G12A12B12C12D12E12F
order122222222222344444444446666666121212121212
size111144466121212222444661212122224488444488

36 irreducible representations

dim1111111111112222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D62+ 1+4S3×D4D4⋊6D6D4○D12
kernelC6.382+ 1+4D6⋊D4C23.11D6C4⋊C4⋊7S3C12⋊D4C12⋊7D4C23⋊2D6D6⋊3D4C12⋊3D4C3×C4⋊D4C2×C4○D12C2×S3×D4C4⋊D4C4×S3C22⋊C4C4⋊C4C22×C4C2×D4C6C4C2C2
# reps1221112121111421132222

Matrix representation of C6.382+ 1+4 ►in GL8(𝔽13)

120000000
012000000
000120000
001120000
00001000
00000100
00000010
00000001
,
120000000
012000000
001200000
000120000
00001560
0000101206
000000128
00000031
,
10000000
012000000
001200000
000120000
00001000
00000100
000047120
000019012
,
01000000
10000000
00010000
00100000
00000100
000012000
00009401
000044120
,
01000000
10000000
00100000
00010000
00000100
00001000
00003401
000091010

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,5,12,0,0,0,0,0,0,6,0,12,3,0,0,0,0,0,6,8,1],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,4,1,0,0,0,0,0,1,7,9,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,9,4,0,0,0,0,1,0,4,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,3,9,0,0,0,0,1,0,4,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C6.382+ 1+4 in GAP, Magma, Sage, TeX 

C_6._{38}2_+^{1+4}
 
% in TeX
 
G:=Group("C6.38ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1166);
 
// by ID
 
G=gap.SmallGroup(192,1166);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,570,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=b^2*d>;
 
// generators/relations
 

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