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G = C62.6D4order 288 = 25·32

6th non-split extension by C62 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.6D4, C322C83C4, C3⋊Dic3.2Q8, (C3×C6).6SD16, C323(C4.Q8), C22.10S3≀C2, C62.C22.2C2, C2.3(C322SD16), (C3×C6).4(C4⋊C4), C2.4(C3⋊S3.Q8), C3⋊Dic3.12(C2×C4), (C2×C322C8).8C2, (C2×C3⋊Dic3).4C22, SmallGroup(288,390)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C62.6D4
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3C62.C22 — C62.6D4
C32C3×C6C3⋊Dic3 — C62.6D4
C1C22

Generators and relations for C62.6D4
 G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=a3b4, dad-1=a-1, cbc-1=a2b3, bd=db, dcd-1=c3 >

Subgroups: 264 in 62 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×2], C4 [×4], C22, C6 [×6], C8 [×2], C2×C4 [×3], C32, Dic3 [×6], C12 [×2], C2×C6 [×2], C4⋊C4 [×2], C2×C8, C3×C6, C3×C6 [×2], C2×Dic3 [×4], C2×C12 [×2], C4.Q8, C3×Dic3 [×2], C3⋊Dic3 [×2], C62, Dic3⋊C4 [×2], C322C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22 [×2], C2×C322C8, C62.6D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, SD16 [×2], C4.Q8, S3≀C2, C3⋊S3.Q8, C322SD16 [×2], C62.6D4

Character table of C62.6D4

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E12F12G12H
 size 111144121212121818444444181818181212121212121212
ρ1111111111111111111111111111111    trivial
ρ2111111-111-111111111-1-1-1-1-11-1-1111-1    linear of order 2
ρ31111111-1-1111111111-1-1-1-11-111-1-1-11    linear of order 2
ρ4111111-1-1-1-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-1-1111i-ii-i-11-1-11-11-11-11-1-i-iiiii-i-i    linear of order 4
ρ61-1-1111-i-iii-11-1-11-11-1-11-11i-i-i-iii-ii    linear of order 4
ρ71-1-1111ii-i-i-11-1-11-11-1-11-11-iiii-i-ii-i    linear of order 4
ρ81-1-1111-ii-ii-11-1-11-11-11-11-1ii-i-i-i-iii    linear of order 4
ρ92222220000-2-2222222000000000000    orthogonal lifted from D4
ρ102-2-222200002-2-2-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ1122-2-2220000002-2-22-2-2--2-2-2--200000000    complex lifted from SD16
ρ1222-2-2220000002-2-22-2-2-2--2--2-200000000    complex lifted from SD16
ρ132-22-222000000-22-2-2-22--2--2-2-200000000    complex lifted from SD16
ρ142-22-222000000-22-2-2-22-2-2--2--200000000    complex lifted from SD16
ρ1544441-2-200-200-2-2-2111000010110001    orthogonal lifted from S3≀C2
ρ164444-21022000111-2-2-200000-100-1-1-10    orthogonal lifted from S3≀C2
ρ174444-210-2-2000111-2-2-2000001001110    orthogonal lifted from S3≀C2
ρ1844441-2200200-2-2-21110000-10-1-1000-1    orthogonal lifted from S3≀C2
ρ194-44-4-21000000-11-122-2000003003-3-30    symplectic lifted from C322SD16, Schur index 2
ρ2044-4-41-2000000-2221-1-10000303-3000-3    symplectic lifted from C322SD16, Schur index 2
ρ2144-4-41-2000000-2221-1-10000-30-330003    symplectic lifted from C322SD16, Schur index 2
ρ224-44-4-21000000-11-122-200000-300-3330    symplectic lifted from C322SD16, Schur index 2
ρ234-44-41-20000002-22-1-110000-30--3-3000--3    complex lifted from C322SD16
ρ2444-4-4-210000001-1-1-22200000--300-3--3-30    complex lifted from C322SD16
ρ2544-4-4-210000001-1-1-22200000-300--3-3--30    complex lifted from C322SD16
ρ264-44-41-20000002-22-1-110000--30-3--3000-3    complex lifted from C322SD16
ρ274-4-441-22i00-2i0022-2-11-10000i0-i-i000i    complex lifted from C3⋊S3.Q8
ρ284-4-44-210-2i2i000-1-112-2200000i00-i-ii0    complex lifted from C3⋊S3.Q8
ρ294-4-441-2-2i002i0022-2-11-10000-i0ii000-i    complex lifted from C3⋊S3.Q8
ρ304-4-44-2102i-2i000-1-112-2200000-i00ii-i0    complex lifted from C3⋊S3.Q8

Smallest permutation representation of C62.6D4
On 96 points
Generators in S96
(1 79)(2 16 26 80 52 88)(3 73)(4 82 54 74 28 10)(5 75)(6 12 30 76 56 84)(7 77)(8 86 50 78 32 14)(9 53)(11 55)(13 49)(15 51)(17 89 59 46 35 71)(18 47)(19 65 37 48 61 91)(20 41)(21 93 63 42 39 67)(22 43)(23 69 33 44 57 95)(24 45)(25 87)(27 81)(29 83)(31 85)(34 96)(36 90)(38 92)(40 94)(58 70)(60 72)(62 66)(64 68)
(1 55 25 5 51 29)(2 6)(3 31 53 7 27 49)(4 8)(9 77 81 13 73 85)(10 14)(11 87 75 15 83 79)(12 16)(17 21)(18 64 36 22 60 40)(19 23)(20 34 62 24 38 58)(26 30)(28 32)(33 37)(35 39)(41 96 66 45 92 70)(42 46)(43 72 94 47 68 90)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)
(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 47 79 18)(2 42 80 21)(3 45 73 24)(4 48 74 19)(5 43 75 22)(6 46 76 17)(7 41 77 20)(8 44 78 23)(9 58 53 70)(10 61 54 65)(11 64 55 68)(12 59 56 71)(13 62 49 66)(14 57 50 69)(15 60 51 72)(16 63 52 67)(25 90 87 36)(26 93 88 39)(27 96 81 34)(28 91 82 37)(29 94 83 40)(30 89 84 35)(31 92 85 38)(32 95 86 33)

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

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

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

Matrix representation of C62.6D4 in GL6(𝔽73)

7200000
0720000
0072000
0007200
00451811
0000720
,
7200000
0720000
0007200
0017200
0092710
0092701
,
19600000
34420000
0026254619
0026251946
0047687223
0020687223
,
24160000
5490000
0027000
0002700
0040162727
006641046

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,45,0,0,0,0,72,18,0,0,0,0,0,1,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,9,9,0,0,72,72,27,27,0,0,0,0,1,0,0,0,0,0,0,1],[19,34,0,0,0,0,60,42,0,0,0,0,0,0,26,26,47,20,0,0,25,25,68,68,0,0,46,19,72,72,0,0,19,46,23,23],[24,5,0,0,0,0,16,49,0,0,0,0,0,0,27,0,40,66,0,0,0,27,16,41,0,0,0,0,27,0,0,0,0,0,27,46] >;

C62.6D4 in GAP, Magma, Sage, TeX

C_6^2._6D_4
% in TeX

G:=Group("C6^2.6D4");
// GroupNames label

G:=SmallGroup(288,390);
// by ID

G=gap.SmallGroup(288,390);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,176,422,219,100,2693,2028,691,797,2372]);
// Polycyclic

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

Export

Character table of C62.6D4 in TeX

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