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G = C62.15D4order 288 = 25·32

15th non-split extension by C62 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.15D4, C32⋊Q164C2, C22.6S3≀C2, C3⋊Dic3.33D4, C62.C42C2, D6.4D6.2C2, C322SD162C2, C322(C8.C22), D6⋊S3.3C22, C3⋊Dic3.11C23, C322C8.2C22, C322Q8.7C22, C2.20(C2×S3≀C2), (C3×C6).20(C2×D4), (C2×C322Q8)⋊10C2, (C2×C3⋊Dic3).98C22, SmallGroup(288,887)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C62.15D4
C1C32C3×C6C3⋊Dic3D6⋊S3C322SD16 — C62.15D4
C32C3×C6C3⋊Dic3 — C62.15D4
C1C2C22

Generators and relations for C62.15D4
 G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=c3 >

Subgroups: 464 in 99 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×2], C4 [×5], C22, C22, S3, C6 [×6], C8 [×2], C2×C4 [×3], D4 [×2], Q8 [×4], C32, Dic3 [×7], C12 [×3], D6, C2×C6 [×3], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, Dic6 [×5], C4×S3, C2×Dic3 [×4], C3⋊D4 [×2], C2×C12, C3×D4, C8.C22, C3×Dic3 [×3], C3⋊Dic3 [×2], S3×C6, C62, C2×Dic6, D42S3, C322C8 [×2], S3×Dic3, D6⋊S3, C322Q8, C322Q8 [×2], C322Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C322SD16 [×2], C32⋊Q16 [×2], C62.C4, D6.4D6, C2×C322Q8, C62.15D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C8.C22, S3≀C2, C2×S3≀C2, C62.15D4

Character table of C62.15D4

 class 12A2B2C3A3B4A4B4C4D4E6A6B6C6D6E6F8A8B12A12B12C12D12E
 size 11212441212121818444482436361212121224
ρ1111111111111111111111111    trivial
ρ2111111-1-1111111111-1-1-1-1-1-11    linear of order 2
ρ311-1111-11-11-1-11-11-111-1-111-1-1    linear of order 2
ρ411-11111-1-11-1-11-11-11-111-1-11-1    linear of order 2
ρ511-1-111-1111-1-11-11-1-1-11-111-11    linear of order 2
ρ611-1-1111-111-1-11-11-1-11-11-1-111    linear of order 2
ρ7111-11111-11111111-1-1-11111-1    linear of order 2
ρ8111-111-1-1-11111111-111-1-1-1-1-1    linear of order 2
ρ922-2022000-22-22-22-200000000    orthogonal lifted from D4
ρ10222022000-2-22222200000000    orthogonal lifted from D4
ρ114442-2100200-2-2-211-1000000-1    orthogonal lifted from S3≀C2
ρ1244-4-2-21002002-221-11000000-1    orthogonal lifted from C2×S3≀C2
ρ1344-401-2-22000-11-1-220001-1-110    orthogonal lifted from C2×S3≀C2
ρ14444-2-2100-200-2-2-21110000001    orthogonal lifted from S3≀C2
ρ1544401-222000111-2-2000-1-1-1-10    orthogonal lifted from S3≀C2
ρ1644401-2-2-2000111-2-200011110    orthogonal lifted from S3≀C2
ρ1744-42-2100-2002-221-1-10000001    orthogonal lifted from C2×S3≀C2
ρ1844-401-22-2000-11-1-22000-111-10    orthogonal lifted from C2×S3≀C2
ρ194-40044000000-40-4000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-4001-2000003-1-320000-33-330    symplectic faithful, Schur index 2
ρ214-4001-2000003-1-3200003-33-30    symplectic faithful, Schur index 2
ρ224-4001-200000-3-132000033-3-30    symplectic faithful, Schur index 2
ρ234-4001-200000-3-1320000-3-3330    symplectic faithful, Schur index 2
ρ248-800-4200000040-2000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C62.15D4
On 48 points
Generators in S48
(1 29 11 37 24 46)(2 34)(3 48 18 39 13 31)(4 36)(5 25 15 33 20 42)(6 38)(7 44 22 35 9 27)(8 40)(10 41)(12 43)(14 45)(16 47)(17 26)(19 28)(21 30)(23 32)
(1 5)(2 16 17 6 12 21)(3 7)(4 23 14 8 19 10)(9 13)(11 15)(18 22)(20 24)(25 29)(26 38 43 30 34 47)(27 31)(28 41 36 32 45 40)(33 37)(35 39)(42 46)(44 48)
(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)
(2 4)(3 7)(6 8)(9 18)(10 21)(11 24)(12 19)(13 22)(14 17)(15 20)(16 23)(25 46)(26 41)(27 44)(28 47)(29 42)(30 45)(31 48)(32 43)(33 37)(34 40)(36 38)

G:=sub<Sym(48)| (1,29,11,37,24,46)(2,34)(3,48,18,39,13,31)(4,36)(5,25,15,33,20,42)(6,38)(7,44,22,35,9,27)(8,40)(10,41)(12,43)(14,45)(16,47)(17,26)(19,28)(21,30)(23,32), (1,5)(2,16,17,6,12,21)(3,7)(4,23,14,8,19,10)(9,13)(11,15)(18,22)(20,24)(25,29)(26,38,43,30,34,47)(27,31)(28,41,36,32,45,40)(33,37)(35,39)(42,46)(44,48), (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), (2,4)(3,7)(6,8)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43)(33,37)(34,40)(36,38)>;

G:=Group( (1,29,11,37,24,46)(2,34)(3,48,18,39,13,31)(4,36)(5,25,15,33,20,42)(6,38)(7,44,22,35,9,27)(8,40)(10,41)(12,43)(14,45)(16,47)(17,26)(19,28)(21,30)(23,32), (1,5)(2,16,17,6,12,21)(3,7)(4,23,14,8,19,10)(9,13)(11,15)(18,22)(20,24)(25,29)(26,38,43,30,34,47)(27,31)(28,41,36,32,45,40)(33,37)(35,39)(42,46)(44,48), (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), (2,4)(3,7)(6,8)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43)(33,37)(34,40)(36,38) );

G=PermutationGroup([(1,29,11,37,24,46),(2,34),(3,48,18,39,13,31),(4,36),(5,25,15,33,20,42),(6,38),(7,44,22,35,9,27),(8,40),(10,41),(12,43),(14,45),(16,47),(17,26),(19,28),(21,30),(23,32)], [(1,5),(2,16,17,6,12,21),(3,7),(4,23,14,8,19,10),(9,13),(11,15),(18,22),(20,24),(25,29),(26,38,43,30,34,47),(27,31),(28,41,36,32,45,40),(33,37),(35,39),(42,46),(44,48)], [(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)], [(2,4),(3,7),(6,8),(9,18),(10,21),(11,24),(12,19),(13,22),(14,17),(15,20),(16,23),(25,46),(26,41),(27,44),(28,47),(29,42),(30,45),(31,48),(32,43),(33,37),(34,40),(36,38)])

Matrix representation of C62.15D4 in GL8(𝔽73)

7272000000
10000000
00100000
00010000
0000461900
0000272700
0000004619
0000002727
,
10000000
01000000
0072720000
00100000
000072000
000007200
000000720
000000072
,
00100000
00010000
10000000
7272000000
0000007271
00000001
000072000
00001100
,
10000000
7272000000
00100000
00010000
00001000
0000727200
0000007271
00000001

G:=sub<GL(8,GF(73))| [72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,19,27,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,19,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,71,1,0,0],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,71,1] >;

C62.15D4 in GAP, Magma, Sage, TeX

C_6^2._{15}D_4
% in TeX

G:=Group("C6^2.15D4");
// GroupNames label

G:=SmallGroup(288,887);
// by ID

G=gap.SmallGroup(288,887);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,675,346,80,2693,2028,362,797,1203]);
// Polycyclic

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

Export

Character table of C62.15D4 in TeX

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