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G = C62.4D4order 288 = 25·32

4th non-split extension by C62 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.4D4, (C3×C6).1Q16, C22.9S3≀C2, C322Q84C4, C3⋊Dic3.5D4, (C3×C6).5SD16, C2.1(C32⋊Q16), C323(Q8⋊C4), C62.C22.1C2, C2.2(C322SD16), C2.10(S32⋊C4), C3⋊Dic3.10(C2×C4), (C2×C322C8).1C2, (C2×C322Q8).1C2, (C3×C6).10(C22⋊C4), (C2×C3⋊Dic3).2C22, SmallGroup(288,388)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C62.4D4
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3C62.C22 — C62.4D4
C32C3×C6C3⋊Dic3 — C62.4D4
C1C22

Generators and relations for C62.4D4
 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=a3c3 >

Subgroups: 328 in 74 conjugacy classes, 19 normal (17 characteristic)
C1, C2 [×3], C3 [×2], C4 [×5], C22, C6 [×6], C8, C2×C4 [×3], Q8 [×3], C32, Dic3 [×7], C12 [×3], C2×C6 [×2], C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], Dic6 [×4], C2×Dic3 [×4], C2×C12 [×2], Q8⋊C4, C3×Dic3 [×3], C3⋊Dic3 [×2], C62, Dic3⋊C4, C2×Dic6, C322C8, C322Q8 [×2], C322Q8, C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22, C2×C322C8, C2×C322Q8, C62.4D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, Q8⋊C4, S3≀C2, S32⋊C4, C322SD16, C32⋊Q16, C62.4D4

Character table of C62.4D4

 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
ρ51-1-1111-i-11i-11-1-11-11-1i-ii-ii-1-i-i11-1i    linear of order 4
ρ61-1-1111-i1-1i-11-1-11-11-1-ii-iii1-i-i-1-11i    linear of order 4
ρ71-1-1111i1-1-i-11-1-11-11-1i-ii-i-i1ii-1-11-i    linear of order 4
ρ81-1-1111i-11-i-11-1-11-11-1-ii-ii-i-1ii11-1-i    linear of order 4
ρ92-2-222200002-2-2-22-22-2000000000000    orthogonal lifted from D4
ρ102222220000-2-2222222000000000000    orthogonal lifted from D4
ρ1122-2-2220000002-2-22-2-22-2-2200000000    symplectic lifted from Q16, Schur index 2
ρ1222-2-2220000002-2-22-2-2-222-200000000    symplectic lifted from Q16, Schur index 2
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
ρ184-4-44-210-22000-1-112-2200000100-1-110    orthogonal lifted from S32⋊C4
ρ1944441-2200200-2-2-21110000-10-1-1000-1    orthogonal lifted from S3≀C2
ρ204-4-44-2102-2000-1-112-2200000-10011-10    orthogonal lifted from S32⋊C4
ρ2144-4-4-210000001-1-1-22200000-3003-330    symplectic lifted from C32⋊Q16, Schur index 2
ρ224-44-4-21000000-11-122-2000003003-3-30    symplectic lifted from C322SD16, Schur index 2
ρ2344-4-41-2000000-2221-1-10000303-3000-3    symplectic lifted from C32⋊Q16, Schur index 2
ρ2444-4-41-2000000-2221-1-10000-30-330003    symplectic lifted from C32⋊Q16, Schur index 2
ρ2544-4-4-210000001-1-1-22200000300-33-30    symplectic lifted from C32⋊Q16, Schur index 2
ρ264-44-4-21000000-11-122-200000-300-3330    symplectic lifted from C322SD16, Schur index 2
ρ274-4-441-22i00-2i0022-2-11-10000i0-i-i000i    complex lifted from S32⋊C4
ρ284-4-441-2-2i002i0022-2-11-10000-i0ii000-i    complex lifted from S32⋊C4
ρ294-44-41-20000002-22-1-110000-30--3-3000--3    complex lifted from C322SD16
ρ304-44-41-20000002-22-1-110000--30-3--3000-3    complex lifted from C322SD16

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

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

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

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

Matrix representation of C62.4D4 in GL6(𝔽73)

100000
010000
0072000
0007200
000011
0000720
,
7200000
0720000
001100
0072000
0000720
0000072
,
40180000
67210000
0000027
0000270
0071400
00596600
,
56550000
16170000
00665900
0014700
0000027
0000270

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[40,67,0,0,0,0,18,21,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,27,0,0,0,0,27,0,0,0],[56,16,0,0,0,0,55,17,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;

C62.4D4 in GAP, Magma, Sage, TeX

C_6^2._4D_4
% in TeX

G:=Group("C6^2.4D4");
// GroupNames label

G:=SmallGroup(288,388);
// by ID

G=gap.SmallGroup(288,388);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,120,422,219,100,2693,2028,691,797,2372]);
// 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=a^3*c^3>;
// generators/relations

Export

Character table of C62.4D4 in TeX

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