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G = C6.482+ 1+4  order 192 = 26·3

48th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.482+ 1+4, C4⋊C4⋊9D6, (C2×D4)⋊11D6, C4⋊D4⋊23S3, C22⋊C4⋊13D6, (C22×C4)⋊24D6, C23⋊2D6⋊14C2, D6⋊3D4⋊28C2, Dic3⋊D4⋊23C2, D6⋊D4⋊14C2, C12⋊7D4⋊45C2, C12⋊3D4⋊20C2, D6⋊C4⋊32C22, (C6×D4)⋊16C22, D6.D4⋊14C2, C2.32(D4○D12), (C2×C12).46C23, (C2×C6).164C24, C4⋊Dic3⋊13C22, C23.14D6⋊18C2, C2.50(D4⋊6D6), Dic3⋊C4⋊35C22, (C22×C12)⋊31C22, (C4×Dic3)⋊26C22, C23.8D6⋊21C2, C3⋊2(C22.54C24), (C2×D12).146C22, C23.21D6⋊14C2, C6.D4⋊27C22, C23.28D6⋊12C2, (S3×C23).51C22, (C22×S3).71C23, C23.124(C22×S3), C22.185(S3×C23), (C22×C6).192C23, (C2×Dic3).81C23, (C22×Dic3)⋊23C22, (S3×C2×C4)⋊16C22, C4⋊C4⋊S3⋊15C2, (C3×C4⋊D4)⋊26C2, (C3×C4⋊C4)⋊16C22, (C2×C3⋊D4)⋊17C22, (C2×C4).42(C22×S3), (C3×C22⋊C4)⋊18C22, SmallGroup(192,1179)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C6.482+ 1+4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C23⋊2D6 — C6.482+ 1+4
Lower central
C3 — C2×C6 — C6.482+ 1+4
Upper central
C1 — C22 — C4⋊D4

Generators and relations for C6.482+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a3b, be=eb, dcd-1=ece=a3c, ede=a3b2d >

Subgroups: 800 in 252 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C22.54C24, C23.8D6, D6⋊D4, Dic3⋊D4, C23.21D6, D6.D4, C4⋊C4⋊S3, C23.28D6, C12⋊7D4, C23⋊2D6, D6⋊3D4, C23.14D6, C12⋊3D4, C3×C4⋊D4, C6.482+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, C22.54C24, D4⋊6D6, D4○D12, C6.482+ 1+4

Smallest permutation representation of C6.482+ 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 28 7 34)(2 29 8 35)(3 30 9 36)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 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 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(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 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), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (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), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(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)(25,40)(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), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (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), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(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)(25,40)(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)], [(1,28,7,34),(2,29,8,35),(3,30,9,36),(4,25,10,31),(5,26,11,32),(6,27,12,33),(13,37,19,43),(14,38,20,44),(15,39,21,45),(16,40,22,46),(17,41,23,47),(18,42,24,48)], [(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)], [(1,22,10,13),(2,21,11,18),(3,20,12,17),(4,19,7,16),(5,24,8,15),(6,23,9,14),(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),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)]])
 

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E···4I6A6B6C6D6E6F6G12A12B12C12D12E12F
order1222222222344444···46666666121212121212
size11114441212122444412···122224488444488

33 irreducible representations

dim1111111111111122222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D62+ 1+4D4⋊6D6D4○D12
kernelC6.482+ 1+4C23.8D6D6⋊D4Dic3⋊D4C23.21D6D6.D4C4⋊C4⋊S3C23.28D6C12⋊7D4C23⋊2D6D6⋊3D4C23.14D6C12⋊3D4C3×C4⋊D4C4⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C6C2C2
# reps1111111112121112113342

Matrix representation of C6.482+ 1+4 ►in GL8(𝔽13)

1212000000
10000000
000120000
001120000
00001100
000012000
00000001
000000121
,
120880000
0121080000
112100000
21010000
000011900
00004200
00000029
000000411
,
10000000
01000000
1211200000
11120120000
00001000
00000100
0000121120
00001112012
,
24000000
211000000
00920000
001140000
00001055
0000121253
00001112012
0000121120
,
24000000
911000000
001140000
00920000
00001055
00000135
000000120
000000012

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

C6.482+ 1+4 in GAP, Magma, Sage, TeX 

C_6._{48}2_+^{1+4}
 
% in TeX
 
G:=Group("C6.48ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1179);
 
// by ID
 
G=gap.SmallGroup(192,1179);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,1571,570,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=c^2=e^2=1,d^2=a^3*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,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
 
// generators/relations
 

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