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G = C62.121C23order 288 = 25·32

116th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.121C23, Dic32:7C2, C3:Dic3:3D4, C23.18S32, C6.72(S3xD4), (C3xDic3):5D4, C3:3(C12:3D4), C32:4(C4:1D4), (C22xC6).79D6, Dic3:1(C3:D4), (C2xDic3).85D6, (C22xS3).27D6, C2.31(Dic3:D6), (C2xC62).40C22, (C6xDic3).42C22, (C2xC3:D4):6S3, (C6xC3:D4):3C2, C6.67(C2xC3:D4), C2.45(S3xC3:D4), (C2xD6:S3):9C2, C22.144(C2xS32), (C3xC6).167(C2xD4), (C2xC32:7D4):5C2, (S3xC2xC6).49C22, (C2xC3:D12):17C2, (C2xC6).140(C22xS3), (C22xC3:S3).35C22, (C2xC3:Dic3).73C22, SmallGroup(288,627)

Series: Derived Chief Lower central Upper central

C1C62 — C62.121C23
C1C3C32C3xC6C62S3xC2xC6C2xD6:S3 — C62.121C23
C32C62 — C62.121C23
C1C22C23

Generators and relations for C62.121C23
 G = < a,b,c,d,e | a6=b6=c2=d2=e2=1, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=b3c, ece=a3c, ede=a3b3d >

Subgroups: 1090 in 243 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C42, C2xD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22xC6, C4:1D4, C3xDic3, C3:Dic3, S3xC6, C2xC3:S3, C62, C62, C4xDic3, C2xD12, C2xC3:D4, C2xC3:D4, C6xD4, D6:S3, C3:D12, C6xDic3, C3xC3:D4, C2xC3:Dic3, C32:7D4, S3xC2xC6, C22xC3:S3, C2xC62, C12:3D4, Dic32, C2xD6:S3, C2xC3:D12, C6xC3:D4, C2xC32:7D4, C62.121C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C4:1D4, S32, S3xD4, C2xC3:D4, C2xS32, C12:3D4, S3xC3:D4, Dic3:D6, C62.121C23

Smallest permutation representation of C62.121C23
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 18 5 16 3 14)(2 13 6 17 4 15)(7 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(37 47)(38 48)(39 43)(40 44)(41 45)(42 46)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222223334444446···66···6666612121212
size11114121236224666618182···24···41212121212121212

42 irreducible representations

dim111111222222244444
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6C3:D4S32S3xD4C2xS32S3xC3:D4Dic3:D6
kernelC62.121C23Dic32C2xD6:S3C2xC3:D12C6xC3:D4C2xC32:7D4C2xC3:D4C3xDic3C3:Dic3C2xDic3C22xS3C22xC6Dic3C23C6C22C2C2
# reps111221242222814142

Matrix representation of C62.121C23 in GL8(Z)

10000000
01000000
00-100000
000-10000
00001000
00000100
000000-11
000000-10
,
-10000000
0-1000000
00-100000
000-10000
0000-1100
0000-1000
00000010
00000001
,
-1-2000000
01000000
00100000
00-1-10000
00000-100
0000-1000
00000010
00000001
,
-10000000
11000000
00120000
000-10000
0000-1000
00000-100
00000001
00000010
,
-1-2000000
01000000
00-1-20000
00010000
00001000
00000100
00000010
00000001

G:=sub<GL(8,Integers())| [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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-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,-1,0,0,0,0,0,0,0,0,-1,-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],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C62.121C23 in GAP, Magma, Sage, TeX

C_6^2._{121}C_2^3
% in TeX

G:=Group("C6^2.121C2^3");
// GroupNames label

G:=SmallGroup(288,627);
// by ID

G=gap.SmallGroup(288,627);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,303,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d=b^3*c,e*c*e=a^3*c,e*d*e=a^3*b^3*d>;
// generators/relations

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