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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, (C6×Dic3)⋊3C4, (C2×C6).56D12, (C2×C6).8Dic6, C6.5(C4×Dic3), C6.40(D6⋊C4), (C3×C6).10C42, C62.33(C2×C4), C6.5(C4⋊Dic3), (C2×Dic3)⋊2Dic3, C2.2(D6⋊Dic3), C6.9(Dic3⋊C4), (C22×C6).110D6, (C2×C62).6C22, C31(C6.C42), (C22×Dic3).2S3, C22.12(S3×Dic3), C2.2(Dic3⋊Dic3), C2.2(C6.D12), C6.10(C6.D4), C324(C2.C42), C2.2(C62.C22), C22.3(C322Q8), C22.18(C3⋊D12), C22.11(D6⋊S3), C22.11(C6.D6), (C2×C6).69(C4×S3), (C2×C3⋊Dic3)⋊4C4, (Dic3×C2×C6).1C2, (C3×C6).26(C4⋊C4), (C2×C6).53(C3⋊D4), (C2×C6).16(C2×Dic3), (C3×C6).38(C22⋊C4), (C22×C3⋊Dic3).1C2, SmallGroup(288,227)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.6Q8
C1C3C32C3×C6C62C2×C62Dic3×C2×C6 — C62.6Q8
C32C3×C6 — 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 [×3], C2 [×4], C3 [×2], C3, C4 [×6], C22 [×3], C22 [×4], C6 [×14], C6 [×7], C2×C4 [×12], C23, C32, Dic3 [×12], C12 [×4], C2×C6 [×14], C2×C6 [×7], C22×C4 [×3], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×4], C2×Dic3 [×18], C2×C12 [×8], C22×C6 [×2], C22×C6, C2.C42, C3×Dic3 [×4], C3⋊Dic3 [×2], C62 [×3], C62 [×4], C22×Dic3 [×2], C22×Dic3 [×3], C22×C12 [×2], C6×Dic3 [×4], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C62, C6.C42 [×2], Dic3×C2×C6 [×2], C22×C3⋊Dic3, C62.6Q8
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×2], C2×C4 [×3], D4 [×3], Q8, Dic3 [×4], D6 [×2], C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2.C42, S32, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×2], S3×Dic3 [×2], C6.D6, D6⋊S3, C3⋊D12 [×2], C322Q8, C6.C42 [×2], Dic32, D6⋊Dic3 [×2], C6.D12, Dic3⋊Dic3 [×2], 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 23 5 21 3 19)(2 24 6 22 4 20)(7 26 11 30 9 28)(8 27 12 25 10 29)(13 90 17 88 15 86)(14 85 18 89 16 87)(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 10 43)(2 42 11 48)(3 41 12 47)(4 40 7 46)(5 39 8 45)(6 38 9 44)(13 58 80 75)(14 57 81 74)(15 56 82 73)(16 55 83 78)(17 60 84 77)(18 59 79 76)(19 50 25 31)(20 49 26 36)(21 54 27 35)(22 53 28 34)(23 52 29 33)(24 51 30 32)(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 58 21 62)(8 59 22 63)(9 60 23 64)(10 55 24 65)(11 56 19 66)(12 57 20 61)(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), (1,23,5,21,3,19)(2,24,6,22,4,20)(7,26,11,30,9,28)(8,27,12,25,10,29)(13,90,17,88,15,86)(14,85,18,89,16,87)(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), (1,37,10,43)(2,42,11,48)(3,41,12,47)(4,40,7,46)(5,39,8,45)(6,38,9,44)(13,58,80,75)(14,57,81,74)(15,56,82,73)(16,55,83,78)(17,60,84,77)(18,59,79,76)(19,50,25,31)(20,49,26,36)(21,54,27,35)(22,53,28,34)(23,52,29,33)(24,51,30,32)(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)(7,58,21,62)(8,59,22,63)(9,60,23,64)(10,55,24,65)(11,56,19,66)(12,57,20,61)(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), (1,23,5,21,3,19)(2,24,6,22,4,20)(7,26,11,30,9,28)(8,27,12,25,10,29)(13,90,17,88,15,86)(14,85,18,89,16,87)(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), (1,37,10,43)(2,42,11,48)(3,41,12,47)(4,40,7,46)(5,39,8,45)(6,38,9,44)(13,58,80,75)(14,57,81,74)(15,56,82,73)(16,55,83,78)(17,60,84,77)(18,59,79,76)(19,50,25,31)(20,49,26,36)(21,54,27,35)(22,53,28,34)(23,52,29,33)(24,51,30,32)(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)(7,58,21,62)(8,59,22,63)(9,60,23,64)(10,55,24,65)(11,56,19,66)(12,57,20,61)(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)], [(1,23,5,21,3,19),(2,24,6,22,4,20),(7,26,11,30,9,28),(8,27,12,25,10,29),(13,90,17,88,15,86),(14,85,18,89,16,87),(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)], [(1,37,10,43),(2,42,11,48),(3,41,12,47),(4,40,7,46),(5,39,8,45),(6,38,9,44),(13,58,80,75),(14,57,81,74),(15,56,82,73),(16,55,83,78),(17,60,84,77),(18,59,79,76),(19,50,25,31),(20,49,26,36),(21,54,27,35),(22,53,28,34),(23,52,29,33),(24,51,30,32),(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),(7,58,21,62),(8,59,22,63),(9,60,23,64),(10,55,24,65),(11,56,19,66),(12,57,20,61),(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+++++--+-++-+-+-
imageC1C2C2C4C4S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4S32S3×Dic3C6.D6D6⋊S3C3⋊D12C322Q8
kernelC62.6Q8Dic3×C2×C6C22×C3⋊Dic3C6×Dic3C2×C3⋊Dic3C22×Dic3C62C62C2×Dic3C22×C6C2×C6C2×C6C2×C6C2×C6C23C22C22C22C22C22
# 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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