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G = C62.19C23order 288 = 25·32

14th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.19C23, Dic3212C2, C12.36(C4×S3), C4⋊Dic311S3, (C2×C12).129D6, C6.2(D42S3), (C6×C12).89C22, (C2×Dic3).56D6, C2.2(D12⋊S3), C6.22(Q83S3), C324(C42⋊C2), C6.D12.6C2, C4.10(C6.D6), (C6×Dic3).54C22, (C4×C3⋊S3)⋊1C4, C6.30(S3×C2×C4), (C2×C4).112S32, (C3×C4⋊Dic3)⋊6C2, C22.22(C2×S32), C32(C4⋊C47S3), (C3×C12).64(C2×C4), (C3×C6).8(C4○D4), C2.8(C2×C6.D6), C3⋊Dic3.43(C2×C4), (C2×C6).38(C22×S3), (C3×C6).50(C22×C4), (C22×C3⋊S3).63C22, (C2×C3⋊Dic3).115C22, (C2×C4×C3⋊S3).2C2, (C2×C3⋊S3).37(C2×C4), SmallGroup(288,497)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.19C23
C1C3C32C3×C6C62C6×Dic3Dic32 — C62.19C23
C32C3×C6 — C62.19C23
C1C22C2×C4

Generators and relations for C62.19C23
 G = < a,b,c,d,e | a6=b6=1, c2=a3, d2=a3b3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 658 in 179 conjugacy classes, 60 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C2×C4, C2×C4 [×9], C23, C32, Dic3 [×10], C12 [×4], C12 [×6], D6 [×14], C2×C6 [×2], C2×C6, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×12], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C42⋊C2, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C3×C4⋊C4 [×2], S3×C2×C4 [×3], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4⋊C47S3 [×2], Dic32 [×2], C6.D12 [×2], C3×C4⋊Dic3 [×2], C2×C4×C3⋊S3, C62.19C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4○D4 [×2], C4×S3 [×4], C22×S3 [×2], C42⋊C2, S32, S3×C2×C4 [×2], D42S3 [×2], Q83S3 [×2], C6.D6 [×2], C2×S32, C4⋊C47S3 [×2], D12⋊S3 [×2], C2×C6.D6, C62.19C23

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

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C···4J4K4L4M4N6A···6F6G6H6I12A···12H12I···12P
order122222333444···444446···666612···1212···12
size11111818224226···699992···24444···412···12

48 irreducible representations

dim11111122222444444
type+++++++++-+++
imageC1C2C2C2C2C4S3D6D6C4○D4C4×S3S32D42S3Q83S3C6.D6C2×S32D12⋊S3
kernelC62.19C23Dic32C6.D12C3×C4⋊Dic3C2×C4×C3⋊S3C4×C3⋊S3C4⋊Dic3C2×Dic3C2×C12C3×C6C12C2×C4C6C6C4C22C2
# reps12221824248122214

Matrix representation of C62.19C23 in GL8(𝔽13)

10000000
01000000
00100000
00010000
000012000
000001200
00000001
0000001212
,
121000000
120000000
001200000
000120000
000012000
000001200
00000010
00000001
,
012000000
120000000
00830000
00550000
00000800
00008000
000000120
000000012
,
120000000
012000000
0012110000
00110000
00000100
00001000
00000010
0000001212
,
10000000
01000000
00500000
00880000
00008000
00000500
000000120
000000012

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

C62.19C23 in GAP, Magma, Sage, TeX

C_6^2._{19}C_2^3
% in TeX

G:=Group("C6^2.19C2^3");
// GroupNames label

G:=SmallGroup(288,497);
// by ID

G=gap.SmallGroup(288,497);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,176,422,219,100,1356,9414]);
// Polycyclic

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

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