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G = C6.622+ 1+4order 192 = 26·3

62nd non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.622+ 1+4, C4⋊C431D6, (C22×C4)⋊9D6, C22⋊C418D6, (C2×D4).99D6, D6⋊Q830C2, C232D6.3C2, D6.22(C4○D4), C23.9D632C2, (C2×C6).203C24, C4⋊Dic340C22, C22.D48S3, C23.12D621C2, C2.64(D46D6), (C2×C12).599C23, Dic3⋊C466C22, D6⋊C4.131C22, (C22×C12)⋊39C22, (C4×Dic3)⋊56C22, (C2×Dic6)⋊31C22, (C6×D4).141C22, C23.8D631C2, C37(C22.45C24), C6.D454C22, (S3×C23).60C22, (C22×C6).223C23, C23.140(C22×S3), C22.224(S3×C23), (C22×S3).209C23, (C2×Dic3).105C23, (C4×C3⋊D4)⋊48C2, C4⋊C47S333C2, C2.65(S3×C4○D4), (C3×C4⋊C4)⋊29C22, (S3×C22⋊C4)⋊15C2, C6.177(C2×C4○D4), (S3×C2×C4).112C22, (C2×C4).65(C22×S3), (C3×C22⋊C4)⋊25C22, (C3×C22.D4)⋊11C2, (C2×C3⋊D4).135C22, SmallGroup(192,1218)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C6.622+ 1+4
C1C3C6C2×C6C22×S3S3×C23S3×C22⋊C4 — C6.622+ 1+4
C3C2×C6 — C6.622+ 1+4
C1C22C22.D4

Generators and relations for C6.622+ 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, dbd-1=ebe=a3b, dcd-1=ece=a3c, ede=b2d >

Subgroups: 672 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×11], C22, C22 [×18], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], Dic3 [×6], C12 [×5], D6 [×4], D6 [×8], C2×C6, C2×C6 [×6], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic6, C4×S3 [×6], C2×Dic3 [×6], C3⋊D4 [×4], C2×C12, C2×C12 [×4], C2×C12, C3×D4, C22×S3 [×2], C22×S3 [×5], C22×C6 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C422C2 [×2], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4, C6.D4 [×4], C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×4], C2×C3⋊D4 [×2], C22×C12, C6×D4, S3×C23, C22.45C24, C23.8D6 [×2], S3×C22⋊C4 [×2], C23.9D6 [×2], C4⋊C47S3 [×2], D6⋊Q8 [×2], C4×C3⋊D4 [×2], C23.12D6, C232D6, C3×C22.D4, C6.622+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ 1+4, S3×C23, C22.45C24, D46D6, S3×C4○D4 [×2], C6.622+ 1+4

Smallest permutation representation of C6.622+ 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 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 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), (1,22,10,13)(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)(25,37)(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), (1,22,10,13)(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)(25,37)(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)], [(1,22,10,13),(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),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)])

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O6A6B6C6D6E6F12A12B12C12D12E12F12G
order1222222222344444444444444466666612121212121212
size1111446666222224446666121212122224484444888

39 irreducible representations

dim1111111111222222444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4○D42+ 1+4D46D6S3×C4○D4
kernelC6.622+ 1+4C23.8D6S3×C22⋊C4C23.9D6C4⋊C47S3D6⋊Q8C4×C3⋊D4C23.12D6C232D6C3×C22.D4C22.D4C22⋊C4C4⋊C4C22×C4C2×D4D6C6C2C2
# reps1222222111132118124

Matrix representation of C6.622+ 1+4 in GL6(𝔽13)

1200000
0120000
00121200
001000
0000120
0000012
,
080000
500000
001000
000100
000080
000005
,
100000
0120000
001000
000100
000010
0000012
,
010000
100000
001000
00121200
000001
0000120
,
010000
100000
001000
000100
000001
000010

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

C6.622+ 1+4 in GAP, Magma, Sage, TeX

C_6._{62}2_+^{1+4}
% in TeX

G:=Group("C6.62ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1218);
// by ID

G=gap.SmallGroup(192,1218);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,570,6278]);
// 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,d*b*d^-1=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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