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G = C62.6Q8order 288 = 25·32

3rd non-split extension by C62 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.6Q8, C62.30D4, C2.5Dic32, C23.38S32, (C6xDic3):3C4, (C2xC6).56D12, (C2xC6).8Dic6, C6.5(C4xDic3), C6.40(D6:C4), (C3xC6).10C42, C62.33(C2xC4), C6.5(C4:Dic3), (C2xDic3):2Dic3, C2.2(D6:Dic3), C6.9(Dic3:C4), (C22xC6).110D6, (C2xC62).6C22, C3:1(C6.C42), (C22xDic3).2S3, C22.12(S3xDic3), C2.2(Dic3:Dic3), C2.2(C6.D12), C6.10(C6.D4), C32:4(C2.C42), C2.2(C62.C22), C22.3(C32:2Q8), C22.18(C3:D12), C22.11(D6:S3), C22.11(C6.D6), (C2xC6).69(C4xS3), (C2xC3:Dic3):4C4, (Dic3xC2xC6).1C2, (C3xC6).26(C4:C4), (C2xC6).53(C3:D4), (C2xC6).16(C2xDic3), (C3xC6).38(C22:C4), (C22xC3:Dic3).1C2, SmallGroup(288,227)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C62.6Q8
C1C3C32C3xC6C62C2xC62Dic3xC2xC6 — C62.6Q8
C32C3xC6 — C62.6Q8
C1C23

Generators and relations for C62.6Q8
 G = < a,b,c,d | a6=b6=c4=1, d2=a3b3c2, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=a3c-1 >

Subgroups: 530 in 179 conjugacy classes, 80 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2xC4, C23, C32, Dic3, C12, C2xC6, C2xC6, C22xC4, C3xC6, C3xC6, C2xDic3, C2xDic3, C2xC12, C22xC6, C22xC6, C2.C42, C3xDic3, C3:Dic3, C62, C62, C22xDic3, C22xDic3, C22xC12, C6xDic3, C6xDic3, C2xC3:Dic3, C2xC3:Dic3, C2xC62, C6.C42, Dic3xC2xC6, C22xC3:Dic3, C62.6Q8
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, Dic3, D6, C42, C22:C4, C4:C4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2.C42, S32, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, S3xDic3, C6.D6, D6:S3, C3:D12, C32:2Q8, C6.C42, Dic32, D6:Dic3, C6.D12, Dic3:Dic3, C62.C22, C62.6Q8

Smallest permutation representation of C62.6Q8
On 96 points
Generators in S96
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 12 5 10 3 8)(2 7 6 11 4 9)(13 90 17 88 15 86)(14 85 18 89 16 87)(19 26 23 30 21 28)(20 27 24 25 22 29)(31 43 33 45 35 47)(32 44 34 46 36 48)(37 52 39 54 41 50)(38 53 40 49 42 51)(55 72 57 68 59 70)(56 67 58 69 60 71)(61 75 63 77 65 73)(62 76 64 78 66 74)(79 95 83 93 81 91)(80 96 84 94 82 92)
(1 37 22 43)(2 42 23 48)(3 41 24 47)(4 40 19 46)(5 39 20 45)(6 38 21 44)(7 51 30 32)(8 50 25 31)(9 49 26 36)(10 54 27 35)(11 53 28 34)(12 52 29 33)(13 58 80 75)(14 57 81 74)(15 56 82 73)(16 55 83 78)(17 60 84 77)(18 59 79 76)(61 86 67 92)(62 85 68 91)(63 90 69 96)(64 89 70 95)(65 88 71 94)(66 87 72 93)
(1 78 30 71)(2 73 25 72)(3 74 26 67)(4 75 27 68)(5 76 28 69)(6 77 29 70)(7 65 22 55)(8 66 23 56)(9 61 24 57)(10 62 19 58)(11 63 20 59)(12 64 21 60)(13 51 91 43)(14 52 92 44)(15 53 93 45)(16 54 94 46)(17 49 95 47)(18 50 96 48)(31 90 42 79)(32 85 37 80)(33 86 38 81)(34 87 39 82)(35 88 40 83)(36 89 41 84)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,12,5,10,3,8)(2,7,6,11,4,9)(13,90,17,88,15,86)(14,85,18,89,16,87)(19,26,23,30,21,28)(20,27,24,25,22,29)(31,43,33,45,35,47)(32,44,34,46,36,48)(37,52,39,54,41,50)(38,53,40,49,42,51)(55,72,57,68,59,70)(56,67,58,69,60,71)(61,75,63,77,65,73)(62,76,64,78,66,74)(79,95,83,93,81,91)(80,96,84,94,82,92), (1,37,22,43)(2,42,23,48)(3,41,24,47)(4,40,19,46)(5,39,20,45)(6,38,21,44)(7,51,30,32)(8,50,25,31)(9,49,26,36)(10,54,27,35)(11,53,28,34)(12,52,29,33)(13,58,80,75)(14,57,81,74)(15,56,82,73)(16,55,83,78)(17,60,84,77)(18,59,79,76)(61,86,67,92)(62,85,68,91)(63,90,69,96)(64,89,70,95)(65,88,71,94)(66,87,72,93), (1,78,30,71)(2,73,25,72)(3,74,26,67)(4,75,27,68)(5,76,28,69)(6,77,29,70)(7,65,22,55)(8,66,23,56)(9,61,24,57)(10,62,19,58)(11,63,20,59)(12,64,21,60)(13,51,91,43)(14,52,92,44)(15,53,93,45)(16,54,94,46)(17,49,95,47)(18,50,96,48)(31,90,42,79)(32,85,37,80)(33,86,38,81)(34,87,39,82)(35,88,40,83)(36,89,41,84)>;

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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,12,5,10,3,8)(2,7,6,11,4,9)(13,90,17,88,15,86)(14,85,18,89,16,87)(19,26,23,30,21,28)(20,27,24,25,22,29)(31,43,33,45,35,47)(32,44,34,46,36,48)(37,52,39,54,41,50)(38,53,40,49,42,51)(55,72,57,68,59,70)(56,67,58,69,60,71)(61,75,63,77,65,73)(62,76,64,78,66,74)(79,95,83,93,81,91)(80,96,84,94,82,92), (1,37,22,43)(2,42,23,48)(3,41,24,47)(4,40,19,46)(5,39,20,45)(6,38,21,44)(7,51,30,32)(8,50,25,31)(9,49,26,36)(10,54,27,35)(11,53,28,34)(12,52,29,33)(13,58,80,75)(14,57,81,74)(15,56,82,73)(16,55,83,78)(17,60,84,77)(18,59,79,76)(61,86,67,92)(62,85,68,91)(63,90,69,96)(64,89,70,95)(65,88,71,94)(66,87,72,93), (1,78,30,71)(2,73,25,72)(3,74,26,67)(4,75,27,68)(5,76,28,69)(6,77,29,70)(7,65,22,55)(8,66,23,56)(9,61,24,57)(10,62,19,58)(11,63,20,59)(12,64,21,60)(13,51,91,43)(14,52,92,44)(15,53,93,45)(16,54,94,46)(17,49,95,47)(18,50,96,48)(31,90,42,79)(32,85,37,80)(33,86,38,81)(34,87,39,82)(35,88,40,83)(36,89,41,84) );

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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,12,5,10,3,8),(2,7,6,11,4,9),(13,90,17,88,15,86),(14,85,18,89,16,87),(19,26,23,30,21,28),(20,27,24,25,22,29),(31,43,33,45,35,47),(32,44,34,46,36,48),(37,52,39,54,41,50),(38,53,40,49,42,51),(55,72,57,68,59,70),(56,67,58,69,60,71),(61,75,63,77,65,73),(62,76,64,78,66,74),(79,95,83,93,81,91),(80,96,84,94,82,92)], [(1,37,22,43),(2,42,23,48),(3,41,24,47),(4,40,19,46),(5,39,20,45),(6,38,21,44),(7,51,30,32),(8,50,25,31),(9,49,26,36),(10,54,27,35),(11,53,28,34),(12,52,29,33),(13,58,80,75),(14,57,81,74),(15,56,82,73),(16,55,83,78),(17,60,84,77),(18,59,79,76),(61,86,67,92),(62,85,68,91),(63,90,69,96),(64,89,70,95),(65,88,71,94),(66,87,72,93)], [(1,78,30,71),(2,73,25,72),(3,74,26,67),(4,75,27,68),(5,76,28,69),(6,77,29,70),(7,65,22,55),(8,66,23,56),(9,61,24,57),(10,62,19,58),(11,63,20,59),(12,64,21,60),(13,51,91,43),(14,52,92,44),(15,53,93,45),(16,54,94,46),(17,49,95,47),(18,50,96,48),(31,90,42,79),(32,85,37,80),(33,86,38,81),(34,87,39,82),(35,88,40,83),(36,89,41,84)]])

60 conjugacy classes

class 1 2A···2G3A3B3C4A···4H4I4J4K4L6A···6N6O···6U12A···12P
order12···23334···444446···66···612···12
size11···12246···6181818182···24···46···6

60 irreducible representations

dim11111222222222444444
type+++++--+-++-+-+-
imageC1C2C2C4C4S3D4Q8Dic3D6Dic6C4xS3D12C3:D4S32S3xDic3C6.D6D6:S3C3:D12C32:2Q8
kernelC62.6Q8Dic3xC2xC6C22xC3:Dic3C6xDic3C2xC3:Dic3C22xDic3C62C62C2xDic3C22xC6C2xC6C2xC6C2xC6C2xC6C23C22C22C22C22C22
# reps12184231424848121121

Matrix representation of C62.6Q8 in GL8(F13)

10000000
01000000
00100000
00010000
000012000
000001200
00000001
0000001212
,
10000000
01000000
000120000
00110000
00000100
0000121200
00000010
00000001
,
50000000
88000000
00100000
00010000
00005000
00000500
00000010
0000001212
,
1211000000
11000000
00800000
00550000
000012000
00001100
00000010
00000001

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

C62.6Q8 in GAP, Magma, Sage, TeX

C_6^2._6Q_8
% in TeX

G:=Group("C6^2.6Q8");
// GroupNames label

G:=SmallGroup(288,227);
// by ID

G=gap.SmallGroup(288,227);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,36,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=a^3*b^3*c^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^3*c^-1>;
// generators/relations

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