metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.17D4, C23.12D6, (C2×D4).6S3, (C6×D4).5C2, (C2×C4).50D6, C6.48(C2×D4), (C4×Dic3)⋊5C2, C3⋊3(C4.4D4), C4.7(C3⋊D4), (C2×Dic6)⋊10C2, C6.30(C4○D4), C6.D4⋊9C2, (C2×C6).51C23, (C2×C12).33C22, C2.16(D4⋊2S3), C22.58(C22×S3), (C22×C6).19C22, (C2×Dic3).18C22, C2.12(C2×C3⋊D4), SmallGroup(96,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.12D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, dad-1=ab=ba, eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 162 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C4.4D4, C4×Dic3, C6.D4, C2×Dic6, C6×D4, C23.12D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, D4⋊2S3, C2×C3⋊D4, C23.12D6
Character table of C23.12D6
class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
ρ14 | 2 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | -1 | 1 | complex lifted from C3⋊D4 |
ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | -1 | 1 | complex lifted from C3⋊D4 |
ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | √-3 | -√-3 | -√-3 | 1 | -1 | complex lifted from C3⋊D4 |
ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | -√-3 | √-3 | √-3 | 1 | -1 | complex lifted from C3⋊D4 |
ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
(2 8)(4 10)(6 12)(13 26)(14 33)(15 28)(16 35)(17 30)(18 25)(19 32)(20 27)(21 34)(22 29)(23 36)(24 31)(37 43)(39 45)(41 47)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)
(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)
(1 13 40 32)(2 18 41 25)(3 23 42 30)(4 16 43 35)(5 21 44 28)(6 14 45 33)(7 19 46 26)(8 24 47 31)(9 17 48 36)(10 22 37 29)(11 15 38 34)(12 20 39 27)
G:=sub<Sym(48)| (2,8)(4,10)(6,12)(13,26)(14,33)(15,28)(16,35)(17,30)(18,25)(19,32)(20,27)(21,34)(22,29)(23,36)(24,31)(37,43)(39,45)(41,47), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (1,13,40,32)(2,18,41,25)(3,23,42,30)(4,16,43,35)(5,21,44,28)(6,14,45,33)(7,19,46,26)(8,24,47,31)(9,17,48,36)(10,22,37,29)(11,15,38,34)(12,20,39,27)>;
G:=Group( (2,8)(4,10)(6,12)(13,26)(14,33)(15,28)(16,35)(17,30)(18,25)(19,32)(20,27)(21,34)(22,29)(23,36)(24,31)(37,43)(39,45)(41,47), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (1,13,40,32)(2,18,41,25)(3,23,42,30)(4,16,43,35)(5,21,44,28)(6,14,45,33)(7,19,46,26)(8,24,47,31)(9,17,48,36)(10,22,37,29)(11,15,38,34)(12,20,39,27) );
G=PermutationGroup([[(2,8),(4,10),(6,12),(13,26),(14,33),(15,28),(16,35),(17,30),(18,25),(19,32),(20,27),(21,34),(22,29),(23,36),(24,31),(37,43),(39,45),(41,47)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,46),(2,47),(3,48),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,25)], [(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)], [(1,13,40,32),(2,18,41,25),(3,23,42,30),(4,16,43,35),(5,21,44,28),(6,14,45,33),(7,19,46,26),(8,24,47,31),(9,17,48,36),(10,22,37,29),(11,15,38,34),(12,20,39,27)]])
C23.12D6 is a maximal subgroup of
C23.D12 M4(2)⋊D6 C12⋊Q8⋊C2 Dic6.D4 (C2×C8).200D6 D12.D4 (C6×D8).C2 C24⋊11D4 C24.22D4 (C3×Q8).D4 C24.31D4 C24.43D4 D12.38D4 2+ 1+4.4S3 C42.106D6 C42.229D6 C42.114D6 C42.115D6 C24.43D6 C24.46D6 C24⋊9D6 Dic6⋊19D4 C4⋊C4.178D6 C6.712- 1+4 D12⋊20D4 C6.422+ 1+4 C6.452+ 1+4 C6.492+ 1+4 C6.812- 1+4 C6.622+ 1+4 C6.652+ 1+4 C42.139D6 S3×C4.4D4 C42.141D6 C42.166D6 C42⋊28D6 C42.238D6 C24.52D6 C24.53D6 C6.1052- 1+4 C6.1072- 1+4 (C2×C12)⋊17D4 C36.17D4 C12.27D12 C62.33C23 C62.101C23 C62.254C23 C60.44D4 C60.88D4 C6.(D4×D5) C60.17D4
C23.12D6 is a maximal quotient of
C24.15D6 C23⋊2Dic6 C24.21D6 C4.(D6⋊C4) (C4×Dic3)⋊9C4 (C2×C12).288D4 C42.62D6 C42.213D6 C12.16D8 C42.72D6 C12.9Q16 C42.77D6 C24.30D6 C24.31D6 C36.17D4 C12.27D12 C62.33C23 C62.101C23 C62.254C23 C60.44D4 C60.88D4 C6.(D4×D5) C60.17D4
Matrix representation of C23.12D6 ►in GL4(𝔽13) generated by
1 | 0 | 0 | 0 |
0 | 12 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 12 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 12 | 0 |
0 | 0 | 0 | 12 |
12 | 0 | 0 | 0 |
0 | 12 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
9 | 0 | 0 | 0 |
0 | 3 | 0 | 0 |
0 | 0 | 0 | 12 |
0 | 0 | 1 | 0 |
0 | 3 | 0 | 0 |
4 | 0 | 0 | 0 |
0 | 0 | 5 | 0 |
0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,0,1,0,0,12,0],[0,4,0,0,3,0,0,0,0,0,5,0,0,0,0,5] >;
C23.12D6 in GAP, Magma, Sage, TeX
C_2^3._{12}D_6
% in TeX
G:=Group("C2^3.12D6");
// GroupNames label
G:=SmallGroup(96,143);
// by ID
G=gap.SmallGroup(96,143);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,506,116,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations
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