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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(S3×D4), D6⋊Dic3⋊26C2, C62⋊5C4⋊4C2, C6.D4⋊5S3, (C22×C6).66D6, C6.62(C4○D12), (C3×Dic3).19D4, (C2×Dic3).82D6, (C22×S3).23D6, C6.49(D4⋊2S3), C3⋊2(C23.12D6), Dic3.7(C3⋊D4), C32⋊10(C4.4D4), (C2×C62).20C22, C3⋊7(C23.11D6), C2.14(D6.4D6), C2.23(D6.3D6), (C6×Dic3).73C22, (C6×C3⋊D4).1C2, (C2×C3⋊D4).2S3, C2.37(S3×C3⋊D4), C6.60(C2×C3⋊D4), C22.132(C2×S32), (C3×C6).147(C2×D4), (S3×C2×C6).40C22, (C3×C6).77(C4○D4), (C2×C32⋊2Q8)⋊13C2, (C3×C6.D4)⋊13C2, (C2×C6).120(C22×S3), (C2×C3⋊Dic3).63C22, SmallGroup(288,607)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.101C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — 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, C2×C4, D4, Q8, C23, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4.4D4, C3×Dic3, C3×Dic3, C3⋊Dic3, S3×C6, C62, C62, C4×Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C6×D4, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.11D6, C23.12D6, Dic32, D6⋊Dic3, C3×C6.D4, C62⋊5C4, C2×C32⋊2Q8, C6×C3⋊D4, C62.101C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, S32, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C2×S32, C23.11D6, C23.12D6, D6.3D6, D6.4D6, S3×C3⋊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+++++++++++++++-+-
imageC1C2C2C2C2C2C2S3S3D4D6D6D6C4○D4C3⋊D4C4○D12S32S3×D4D4⋊2S3C2×S32D6.3D6D6.4D6S3×C3⋊D4
kernelC62.101C23Dic32D6⋊Dic3C3×C6.D4C62⋊5C4C2×C32⋊2Q8C6×C3⋊D4C6.D4C2×C3⋊D4C3×Dic3C2×Dic3C22×S3C22×C6C3×C6Dic3C6C23C6C6C22C2C2C2
# 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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