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G = C62.7D4  order 288 = 25·32

7th non-split extension by C62 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.7D4, (C3×C6).5D8, (C3×C6).2Q16, C32⋊2C8⋊1C4, C3⋊Dic3.3Q8, C2.2(C32⋊D8), C32⋊2(C2.D8), C22.11S3≀C2, C2.2(C32⋊Q16), C62.C22.3C2, (C3×C6).5(C4⋊C4), C2.5(C3⋊S3.Q8), C3⋊Dic3.13(C2×C4), (C2×C32⋊2C8).5C2, (C2×C3⋊Dic3).5C22, SmallGroup(288,391)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C62.7D4
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C62.C22 — C62.7D4
Lower central
C32 — C3×C6 — C3⋊Dic3 — C62.7D4
Upper central
C1 — C22

Generators and relations for C62.7D4
 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=b3c3 >

Subgroups: 264 in 62 conjugacy classes, 19 normal (15 characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C2×C8, C3×C6, C2×Dic3, C2×C12, C2.D8, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C32⋊2C8, C6×Dic3, C2×C3⋊Dic3, C62.C22, C2×C32⋊2C8, C62.7D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, D8, Q16, C2.D8, S3≀C2, C3⋊S3.Q8, C32⋊D8, C32⋊Q16, C62.7D4

Character table of C62.7D4

 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
ρ102-22-222000000-22-2-2-22√2√2-√2-√200000000    orthogonal lifted from D8
ρ112-22-222000000-22-2-2-22-√2-√2√2√200000000    orthogonal lifted from D8
ρ122-2-222200002-2-2-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ1322-2-2220000002-2-22-2-2√2-√2-√2√200000000    symplectic lifted from Q16, Schur index 2
ρ1422-2-2220000002-2-22-2-2-√2√2√2-√200000000    symplectic lifted from Q16, Schur index 2
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
ρ1944-4-4-210000001-1-1-22200000-√300√3-√3√30    symplectic lifted from C32⋊Q16, Schur index 2
ρ2044-4-41-2000000-2221-1-10000√30√3-√3000-√3    symplectic lifted from C32⋊Q16, Schur index 2
ρ2144-4-41-2000000-2221-1-10000-√30-√3√3000√3    symplectic lifted from C32⋊Q16, Schur index 2
ρ2244-4-4-210000001-1-1-22200000√300-√3√3-√30    symplectic lifted from C32⋊Q16, Schur index 2
ρ234-44-41-20000002-22-1-110000√-30-√-3√-3000-√-3    complex lifted from C32⋊D8
ρ244-44-41-20000002-22-1-110000-√-30√-3-√-3000√-3    complex lifted from C32⋊D8
ρ254-44-4-21000000-11-122-200000-√-300-√-3√-3√-30    complex lifted from C32⋊D8
ρ264-44-4-21000000-11-122-200000√-300√-3-√-3-√-30    complex lifted from C32⋊D8
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 C62.7D4
►On 96 points
Generators in S96
(1 16)(2 78 30 9 51 81)(3 10)(4 83 53 11 32 80)(5 12)(6 74 26 13 55 85)(7 14)(8 87 49 15 28 76)(17 89 62 39 42 67)(18 40)(19 69 44 33 64 91)(20 34)(21 93 58 35 46 71)(22 36)(23 65 48 37 60 95)(24 38)(25 84)(27 86)(29 88)(31 82)(41 96)(43 90)(45 92)(47 94)(50 77)(52 79)(54 73)(56 75)(57 70)(59 72)(61 66)(63 68)

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

Matrix representation of C62.7D4 ►in GL8(𝔽73)

720000000
072000000
00100000
00010000
00001000
00000100
0000007272
00000010
,
10000000
01000000
007200000
000720000
0000727200
00001000
00000010
00000001
,
4533000000
4728000000
000410000
0016410000
000000714
000000766
00001000
00000100
,
2632000000
4547000000
000320000
001600000
00001000
00000100
000000714
000000766

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[45,47,0,0,0,0,0,0,33,28,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,41,41,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,7,7,0,0,0,0,0,0,14,66,0,0],[26,45,0,0,0,0,0,0,32,47,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,66] >;
 

C62.7D4 in GAP, Magma, Sage, TeX 

C_6^2._7D_4
 
% in TeX
 
G:=Group("C6^2.7D4");
 
// GroupNames label
 
G:=SmallGroup(288,391);
 
// by ID
 
G=gap.SmallGroup(288,391);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,64,422,219,100,2693,2028,691,797,2372]);
 
// 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=b^3*c^3>;
 
// generators/relations
 

Export

Character table of C62.7D4 in TeX

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