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G = C62.101C23  order 288 = 25·32

96th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.101C23, Dic32:4C2, C23.13S32, C6.164(S3xD4), D6:Dic3:26C2, C62:5C4:4C2, C6.D4:5S3, (C22xC6).66D6, C6.62(C4oD12), (C3xDic3).19D4, (C2xDic3).82D6, (C22xS3).23D6, C6.49(D4:2S3), C3:2(C23.12D6), Dic3.7(C3:D4), C32:10(C4.4D4), (C2xC62).20C22, C3:7(C23.11D6), C2.14(D6.4D6), C2.23(D6.3D6), (C6xDic3).73C22, (C6xC3:D4).1C2, (C2xC3:D4).2S3, C2.37(S3xC3:D4), C6.60(C2xC3:D4), C22.132(C2xS32), (C3xC6).147(C2xD4), (S3xC2xC6).40C22, (C3xC6).77(C4oD4), (C2xC32:2Q8):13C2, (C3xC6.D4):13C2, (C2xC6).120(C22xS3), (C2xC3:Dic3).63C22, SmallGroup(288,607)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.101C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — D6:Dic3 — C62.101C23
Lower central
C32 — C62 — C62.101C23
Upper central
C1 — C22 — C23

Generators and relations for C62.101C23
 G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ece=a3b3c, ede=b3d >

Subgroups: 602 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, Q8, C23, C23, C32, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C42, C22:C4, C2xD4, C2xQ8, C3xS3, C3xC6, C3xC6, Dic6, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C4.4D4, C3xDic3, C3xDic3, C3:Dic3, S3xC6, C62, C62, C4xDic3, D6:C4, C6.D4, C6.D4, C3xC22:C4, C2xDic6, C2xC3:D4, C6xD4, C32:2Q8, C6xDic3, C3xC3:D4, C2xC3:Dic3, S3xC2xC6, C2xC62, C23.11D6, C23.12D6, Dic32, D6:Dic3, C3xC6.D4, C62:5C4, C2xC32:2Q8, C6xC3:D4, C62.101C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4.4D4, S32, C4oD12, S3xD4, D4:2S3, C2xC3:D4, C2xS32, C23.11D6, C23.12D6, D6.3D6, D6.4D6, S3xC3:D4, C62.101C23

Smallest permutation representation of C62.101C23
►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 16 3 18 5 14)(2 17 4 13 6 15)(7 46 9 48 11 44)(8 47 10 43 12 45)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 39 35 37 33 41)(32 40 36 38 34 42)

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G···6Q6R6S12A···12F
order122222333444444446···66···66612···12
size11114122246666121818362···24···4121212···12

42 irreducible representations

dim11111112222222224444444
type+++++++++++++++-+-
imageC1C2C2C2C2C2C2S3S3D4D6D6D6C4oD4C3:D4C4oD12S32S3xD4D4:2S3C2xS32D6.3D6D6.4D6S3xC3:D4
kernelC62.101C23Dic32D6:Dic3C3xC6.D4C62:5C4C2xC32:2Q8C6xC3:D4C6.D4C2xC3:D4C3xDic3C2xDic3C22xS3C22xC6C3xC6Dic3C6C23C6C6C22C2C2C2
# reps11211111123124441131222

Matrix representation of C62.101C23 ►in GL8(F13)

10000000
01000000
001200000
000120000
00001000
00000100
000000121
000000120
,
120000000
012000000
001200000
000120000
000012100
000012000
00000010
00000001
,
12000000
012000000
001200000
00210000
00000100
00001000
00000010
00000001
,
510000000
88000000
0012120000
00210000
00001000
00000100
00000001
00000010
,
10000000
1212000000
00100000
0011120000
000012000
000001200
00000010
00000001

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

C62.101C23 in GAP, Magma, Sage, TeX 

C_6^2._{101}C_2^3
 
% in TeX
 
G:=Group("C6^2.101C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,607);
 
// by ID
 
G=gap.SmallGroup(288,607);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=e^2=1,d^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,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,e*c*e=a^3*b^3*c,e*d*e=b^3*d>;
 
// generators/relations
 

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