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G = C6.372+ 1+4  order 192 = 26·3

37th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.372+ 1+4, C4⋊C4⋊5D6, (C2×D4)⋊21D6, C4⋊D4⋊9S3, C22⋊C4⋊9D6, D6.39(C2×D4), (C22×S3)⋊7D4, (C22×C4)⋊18D6, C23⋊2D6⋊22C2, D6⋊D4⋊11C2, C12⋊7D4⋊44C2, C22.3(S3×D4), D6⋊C4⋊66C22, (C6×D4)⋊29C22, (C2×D12)⋊7C22, C3⋊3(C23⋊3D4), C6.64(C22×D4), C23.9D6⋊17C2, D6.D4⋊11C2, C2.26(D4○D12), (C2×C6).149C24, (C2×C12).38C23, (S3×C23)⋊9C22, C4⋊Dic3⋊11C22, C23.14D6⋊12C2, C2.39(D4⋊6D6), Dic3⋊C4⋊15C22, (C22×C12)⋊30C22, (C22×C6).18C23, C6.D4⋊21C22, C22.170(S3×C23), C23.190(C22×S3), (C2×Dic3).70C23, (C22×S3).184C23, (C22×Dic3)⋊18C22, (C2×S3×D4)⋊9C2, C2.37(C2×S3×D4), (C2×C6).5(C2×D4), (C2×D6⋊C4)⋊25C2, (C3×C4⋊C4)⋊8C22, (S3×C2×C4)⋊12C22, (C3×C4⋊D4)⋊11C2, (C2×C3⋊D4)⋊13C22, (C3×C22⋊C4)⋊10C22, (C2×C4).175(C22×S3), SmallGroup(192,1164)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C6.372+ 1+4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C6.372+ 1+4
Lower central
C3 — C2×C6 — C6.372+ 1+4
Upper central
C1 — C22 — C4⋊D4

Generators and relations for C6.372+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 1136 in 346 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C22×D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, S3×C23, C23⋊3D4, D6⋊D4, C23.9D6, D6.D4, C2×D6⋊C4, C12⋊7D4, C23⋊2D6, C23.14D6, C3×C4⋊D4, C2×S3×D4, C6.372+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, C23⋊3D4, C2×S3×D4, D4⋊6D6, D4○D12, C6.372+ 1+4

Smallest permutation representation of C6.372+ 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 46 7 40)(2 47 8 41)(3 48 9 42)(4 43 10 37)(5 44 11 38)(6 45 12 39)(13 34 19 28)(14 35 20 29)(15 36 21 30)(16 31 22 25)(17 32 23 26)(18 33 24 27)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M 3 4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G12A12B12C12D12E12F
order122222222222223444444446666666121212121212
size111122446666121224444121212122224488444488

36 irreducible representations

dim11111111112222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D4D6D6D6D62+ 1+4S3×D4D4⋊6D6D4○D12
kernelC6.372+ 1+4D6⋊D4C23.9D6D6.D4C2×D6⋊C4C12⋊7D4C23⋊2D6C23.14D6C3×C4⋊D4C2×S3×D4C4⋊D4C22×S3C22⋊C4C4⋊C4C22×C4C2×D4C6C22C2C2
# reps12221122121421132222

Matrix representation of C6.372+ 1+4 ►in GL8(𝔽13)

1212000000
10000000
001200000
000120000
000012000
000001200
000000120
000000012
,
120000000
012000000
000120000
001200000
000084116
00001020
000094110
000011964
,
10000000
01000000
00100000
00010000
0000120110
00005927
00000010
000011964
,
10000000
1212000000
000120000
00100000
00005927
0000120110
000048123
00006879
,
10000000
01000000
00010000
001200000
00000100
000012000
000048123
000041081

G:=sub<GL(8,GF(13))| [12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,1,9,11,0,0,0,0,4,0,4,9,0,0,0,0,11,2,1,6,0,0,0,0,6,0,10,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,5,0,11,0,0,0,0,0,9,0,9,0,0,0,0,11,2,1,6,0,0,0,0,0,7,0,4],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,5,12,4,6,0,0,0,0,9,0,8,8,0,0,0,0,2,11,12,7,0,0,0,0,7,0,3,9],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,4,4,0,0,0,0,1,0,8,10,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1] >;
 

C6.372+ 1+4 in GAP, Magma, Sage, TeX 

C_6._{37}2_+^{1+4}
 
% in TeX
 
G:=Group("C6.37ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1164);
 
// by ID
 
G=gap.SmallGroup(192,1164);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,136,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=a^3*b^2,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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