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G = C2×Dic6order 48 = 24·3

Direct product of C2 and Dic6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic6, C6⋊Q8, C4.11D6, C6.1C23, C22.8D6, C12.11C22, Dic3.1C22, C31(C2×Q8), (C2×C4).4S3, (C2×C12).3C2, C2.3(C22×S3), (C2×C6).8C22, (C2×Dic3).3C2, SmallGroup(48,34)

Series: Derived Chief Lower central Upper central

C1C6 — C2×Dic6
C1C3C6Dic3C2×Dic3 — C2×Dic6
C3C6 — C2×Dic6
C1C22C2×C4

Generators and relations for C2×Dic6
 G = < a,b,c | a2=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3C4
3C4
3C2×C4
3Q8
3C2×C4
3Q8
3Q8
3Q8
3C2×Q8

Character table of C2×Dic6

 class 12A2B2C34A4B4C4D4E4F6A6B6C12A12B12C12D
 size 111122266662222222
ρ1111111111111111111    trivial
ρ21-1-111-11-1-111-11-11-11-1    linear of order 2
ρ311111-1-11-11-1111-1-1-1-1    linear of order 2
ρ41-1-1111-11-1-11-11-1-11-11    linear of order 2
ρ51111111-1-1-1-11111111    linear of order 2
ρ61-1-1111-1-111-1-11-1-11-11    linear of order 2
ρ711111-1-1-11-11111-1-1-1-1    linear of order 2
ρ81-1-111-1111-1-1-11-11-11-1    linear of order 2
ρ92-2-22-1-2200001-11-11-11    orthogonal lifted from D6
ρ102222-1220000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ112-2-22-12-200001-111-11-1    orthogonal lifted from D6
ρ122222-1-2-20000-1-1-11111    orthogonal lifted from D6
ρ132-22-220000002-2-20000    symplectic lifted from Q8, Schur index 2
ρ1422-2-22000000-2-220000    symplectic lifted from Q8, Schur index 2
ρ152-22-2-1000000-11133-3-3    symplectic lifted from Dic6, Schur index 2
ρ1622-2-2-100000011-13-3-33    symplectic lifted from Dic6, Schur index 2
ρ172-22-2-1000000-111-3-333    symplectic lifted from Dic6, Schur index 2
ρ1822-2-2-100000011-1-333-3    symplectic lifted from Dic6, Schur index 2

Smallest permutation representation of C2×Dic6
Regular action on 48 points
Generators in S48
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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 34 7 28)(2 33 8 27)(3 32 9 26)(4 31 10 25)(5 30 11 36)(6 29 12 35)(13 39 19 45)(14 38 20 44)(15 37 21 43)(16 48 22 42)(17 47 23 41)(18 46 24 40)

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

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

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

Matrix representation of C2×Dic6 in GL5(𝔽13)

120000
01000
00100
00010
00001
,
10000
04000
001000
000125
000101
,
120000
00100
01000
00062
00017

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,12,10,0,0,0,5,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,2,7] >;

C2×Dic6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_6
% in TeX

G:=Group("C2xDic6");
// GroupNames label

G:=SmallGroup(48,34);
// by ID

G=gap.SmallGroup(48,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,40,182,42,804]);
// Polycyclic

G:=Group<a,b,c|a^2=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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