metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:5D4, C23:2D6, (C2xC4):2D6, (C2xC6):2D4, C3:2C22wrC2, (C2xD4):3S3, (C6xD4):8C2, D6:C4:14C2, C2.25(S3xD4), C6.49(C2xD4), (S3xC23):2C2, (C2xC12):7C22, C22:3(C3:D4), (C2xC6).52C23, (C22xC6):3C22, C6.D4:10C2, (C2xDic3):2C22, C22.59(C22xS3), (C22xS3).25C22, (C2xC3:D4):4C2, C2.13(C2xC3:D4), SmallGroup(96,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23:2D6
G = < a,b,c,d,e | a2=b2=c2=d6=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 354 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22:C4, C2xD4, C2xD4, C24, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22wrC2, D6:C4, C6.D4, C2xC3:D4, C6xD4, S3xC23, C23:2D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C22wrC2, S3xD4, C2xC3:D4, C23:2D6
Character table of C23:2D6
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 2 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ15 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
ρ16 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ18 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from C3:D4 |
ρ20 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | -√-3 | √-3 | √-3 | -√-3 | complex lifted from C3:D4 |
ρ21 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | √-3 | -√-3 | -√-3 | √-3 | complex lifted from C3:D4 |
ρ22 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from C3:D4 |
ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
(1 17)(2 15)(3 13)(4 19)(5 23)(6 21)(7 16)(8 14)(9 18)(10 20)(11 24)(12 22)
(1 4)(2 5)(3 6)(7 11)(8 12)(9 10)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 8)(2 9)(3 7)(4 12)(5 10)(6 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 14)(15 18)(16 17)(19 24)(20 23)(21 22)
G:=sub<Sym(24)| (1,17)(2,15)(3,13)(4,19)(5,23)(6,21)(7,16)(8,14)(9,18)(10,20)(11,24)(12,22), (1,4)(2,5)(3,6)(7,11)(8,12)(9,10)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22)>;
G:=Group( (1,17)(2,15)(3,13)(4,19)(5,23)(6,21)(7,16)(8,14)(9,18)(10,20)(11,24)(12,22), (1,4)(2,5)(3,6)(7,11)(8,12)(9,10)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22) );
G=PermutationGroup([[(1,17),(2,15),(3,13),(4,19),(5,23),(6,21),(7,16),(8,14),(9,18),(10,20),(11,24),(12,22)], [(1,4),(2,5),(3,6),(7,11),(8,12),(9,10),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,8),(2,9),(3,7),(4,12),(5,10),(6,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,14),(15,18),(16,17),(19,24),(20,23),(21,22)]])
G:=TransitiveGroup(24,145);
C23:2D6 is a maximal subgroup of
C3:C2wrC4 C23.3D12 C23:D12 2+ 1+4:7S3 C42:14D6 D12:23D4 C42:18D6 C42:19D6 S3xC22wrC2 C24:7D6 C24:8D6 C24.44D6 C24.45D6 C24:9D6 C6.372+ 1+4 C4:C4:21D6 C6.382+ 1+4 D12:19D4 C6.402+ 1+4 D12:20D4 C6.422+ 1+4 C6.462+ 1+4 C6.482+ 1+4 C6.1202+ 1+4 C4:C4:28D6 C6.612+ 1+4 C6.1222+ 1+4 C6.622+ 1+4 C6.682+ 1+4 C42:22D6 C42:23D6 C42:24D6 C42:28D6 D12:11D4 C42:30D6 D4xC3:D4 C24:12D6 (C2xD4):43D6 C6.1452+ 1+4 C6.1462+ 1+4 C23:2D18 D6:4D12 C62:4D4 C62:8D4 C62.125C23 C62:13D4 D6:S4 D6:4D20 C15:C22wrC2 D30:18D4 D30:8D4 D30:17D4
C23:2D6 is a maximal quotient of
C24.57D6 C23:2Dic6 C24.59D6 C24.23D6 C24.25D6 C23:3D12 (C2xDic3):Q8 D6:C4:6C4 (C2xC4):3D12 C24:6D6 D12:16D4 D12:17D4 Dic6:17D4 D12.36D4 D12.37D4 Dic6.37D4 C22:C4:D6 C42:7D6 D12.14D4 C42:8D6 D12.15D4 D12:D4 Dic6:D4 D6:6SD16 D6:8SD16 D12:7D4 Dic6.16D4 D6:5Q16 D12.17D4 D12:18D4 D12.38D4 D12.39D4 D12.40D4 C24.29D6 C24.32D6 C23:2D18 D6:4D12 C62:4D4 C62:8D4 C62.125C23 C62:13D4 D6:4D20 C15:C22wrC2 D30:18D4 D30:8D4 D30:17D4
Matrix representation of C23:2D6 ►in GL4(F13) generated by
2 | 4 | 0 | 0 |
9 | 11 | 0 | 0 |
0 | 0 | 0 | 3 |
0 | 0 | 9 | 0 |
12 | 0 | 0 | 0 |
0 | 12 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 12 | 0 |
0 | 0 | 0 | 12 |
1 | 1 | 0 | 0 |
12 | 0 | 0 | 0 |
0 | 0 | 12 | 0 |
0 | 0 | 0 | 1 |
1 | 1 | 0 | 0 |
0 | 12 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [2,9,0,0,4,11,0,0,0,0,0,9,0,0,3,0],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,12,0,0,1,0,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,1,12,0,0,0,0,1,0,0,0,0,12] >;
C23:2D6 in GAP, Magma, Sage, TeX
C_2^3\rtimes_2D_6
% in TeX
G:=Group("C2^3:2D6");
// GroupNames label
G:=SmallGroup(96,144);
// by ID
G=gap.SmallGroup(96,144);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^6=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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