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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 [×3], 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 [×3], C2×Dic3 [×4], C2×C12 [×2], C2.D8, C3×Dic3 [×2], C3⋊Dic3 [×2], C62, Dic3⋊C4 [×2], C32⋊2C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22 [×2], C2×C32⋊2C8, C62.7D4
Quotients: C1, C2 [×3], C4 [×2], 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 51)(2 13 30 52 78 81)(3 53)(4 83 80 54 32 15)(5 55)(6 9 26 56 74 85)(7 49)(8 87 76 50 28 11)(10 75)(12 77)(14 79)(16 73)(17 89 33 60 42 67)(18 61)(19 69 44 62 35 91)(20 63)(21 93 37 64 46 71)(22 57)(23 65 48 58 39 95)(24 59)(25 84)(27 86)(29 88)(31 82)(34 68)(36 70)(38 72)(40 66)(41 96)(43 90)(45 92)(47 94)

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

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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