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G = C6.682+ 1+4  order 192 = 26·3

68th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.682+ 1+4, C4⋊C4⋊16D6, D6⋊C4⋊5C22, C22⋊C4⋊19D6, (C22×C4)⋊28D6, Dic3⋊D4⋊35C2, C12⋊D4⋊32C2, C23⋊2D6⋊18C2, D6⋊D4⋊22C2, C12⋊3D4⋊22C2, C12⋊7D4⋊14C2, (C2×D4).105D6, C23.9D6⋊38C2, C2.46(D4○D12), (C2×D12)⋊27C22, (C2×C6).210C24, C4⋊Dic3⋊16C22, C2.70(D4⋊6D6), (C2×C12).184C23, Dic3⋊C4⋊25C22, (C22×C12)⋊13C22, (C4×Dic3)⋊34C22, (C6×D4).148C22, C22.D4⋊15S3, C3⋊3(C22.54C24), (S3×C23).61C22, (C22×S3).91C23, C22.231(S3×C23), C23.141(C22×S3), (C22×C6).224C23, (C2×Dic3).109C23, C6.D4.48C22, (S3×C2×C4)⋊24C22, C4⋊C4⋊S3⋊32C2, (C3×C4⋊C4)⋊30C22, (C2×C3⋊D4)⋊21C22, (C2×C4).71(C22×S3), (C3×C22⋊C4)⋊26C22, (C3×C22.D4)⋊18C2, SmallGroup(192,1225)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C6.682+ 1+4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — D6⋊D4 — C6.682+ 1+4
Lower central
C3 — C2×C6 — C6.682+ 1+4
Upper central
C1 — C22 — C22.D4

Generators and relations for C6.682+ 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=a3b-1, bd=db, ebe=a3b, dcd-1=ece=a3c, ede=b2d >

Subgroups: 832 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22≀C2, C4⋊D4, C22.D4, C22.D4, C42⋊2C2, C4⋊1D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C22.54C24, D6⋊D4, C23.9D6, Dic3⋊D4, C12⋊D4, C4⋊C4⋊S3, C12⋊7D4, C23⋊2D6, C12⋊3D4, C3×C22.D4, C6.682+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, C22.54C24, D4⋊6D6, D4○D12, C6.682+ 1+4

Smallest permutation representation of C6.682+ 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 10 34)(2 26 11 35)(3 27 12 36)(4 28 7 31)(5 29 8 32)(6 30 9 33)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4E4F4G4H4I6A6B6C6D6E6F12A12B12C12D12E12F12G
order122222222234···4444466666612121212121212
size1111441212121224···4121212122224484444888

33 irreducible representations

dim111111111122222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6D62+ 1+4D4⋊6D6D4○D12
kernelC6.682+ 1+4D6⋊D4C23.9D6Dic3⋊D4C12⋊D4C4⋊C4⋊S3C12⋊7D4C23⋊2D6C12⋊3D4C3×C22.D4C22.D4C22⋊C4C4⋊C4C22×C4C2×D4C6C2C2
# reps122222211113211324

Matrix representation of C6.682+ 1+4 ►in GL8(𝔽13)

11000000
120000000
00110000
001200000
00001000
00000100
00000010
00000001
,
001070000
00630000
36000000
710000000
000000120
0000112125
00001000
000031001
,
10000000
01000000
001200000
000120000
00001000
00000100
000000120
0000103012
,
001200000
00110000
120000000
11000000
00000100
000012000
000012118
0000100312
,
00100000
00010000
10000000
01000000
00000100
00001000
000012118
000000012

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

C6.682+ 1+4 in GAP, Magma, Sage, TeX 

C_6._{68}2_+^{1+4}
 
% in TeX
 
G:=Group("C6.68ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1225);
 
// by ID
 
G=gap.SmallGroup(192,1225);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,184,1571,570,6278]);
 
// 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=a^3*b^-1,b*d=d*b,e*b*e=a^3*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=b^2*d>;
 
// generators/relations
 

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