Copied to
clipboard

G = C4⋊D12order 96 = 25·3

The semidirect product of C4 and D12 acting via D12/C12=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C41D12, C124D4, C426S3, (C4×C12)⋊4C2, C6.3(C2×D4), (C2×D12)⋊1C2, C31(C41D4), (C2×C4).76D6, C2.5(C2×D12), (C2×C6).15C23, (C2×C12).87C22, (C22×S3).1C22, C22.36(C22×S3), SmallGroup(96,81)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4⋊D12
C1C3C6C2×C6C22×S3C2×D12 — C4⋊D12
C3C2×C6 — C4⋊D12
C1C22C42

Generators and relations for C4⋊D12
 G = < a,b,c | a4=b12=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 330 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×12], S3 [×4], C6 [×3], C2×C4 [×3], D4 [×12], C23 [×4], C12 [×6], D6 [×12], C2×C6, C42, C2×D4 [×6], D12 [×12], C2×C12 [×3], C22×S3 [×4], C41D4, C4×C12, C2×D12 [×6], C4⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×6], C22×S3, C41D4, C2×D12 [×3], C4⋊D12

Character table of C4⋊D12

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C12A12B12C12D12E12F12G12H12I12J12K12L
 size 1111121212122222222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211111-11-11-1-11-1-11111-1-11-1-11-1-11-1-11    linear of order 2
ρ3111111-1-1111-1-1-1-111111-1-1-1-1-1-1-111-1    linear of order 2
ρ411111-1-111-1-1-111-1111-1-1-111-111-1-1-1-1    linear of order 2
ρ51111-1-111111-1-1-1-111111-1-1-1-1-1-1-111-1    linear of order 2
ρ61111-111-11-1-1-111-1111-1-1-111-111-1-1-1-1    linear of order 2
ρ71111-1-1-1-11111111111111111111111    linear of order 2
ρ81111-11-111-1-11-1-11111-1-11-1-11-1-11-1-11    linear of order 2
ρ92-2-22000020002-202-2-2000220-2-20000    orthogonal lifted from D4
ρ102-22-20000200200-2-2-2200-200-2002002    orthogonal lifted from D4
ρ112-2-2200002000-2202-2-2000-2-20220000    orthogonal lifted from D4
ρ1222220000-1222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1322220000-122-2-2-2-2-1-1-1-1-11111111-1-11    orthogonal lifted from D6
ρ1422220000-1-2-2-222-2-1-1-1111-1-11-1-11111    orthogonal lifted from D6
ρ1522-2-2000022-20000-22-2220000000-2-20    orthogonal lifted from D4
ρ1622-2-200002-220000-22-2-2-20000000220    orthogonal lifted from D4
ρ172-22-20000200-2002-2-2200200200-200-2    orthogonal lifted from D4
ρ1822220000-1-2-22-2-22-1-1-111-111-111-111-1    orthogonal lifted from D6
ρ192-22-20000-100200-211-13-313-31-33-13-3-1    orthogonal lifted from D12
ρ2022-2-20000-1-2200001-111133-3-33-3-3-1-13    orthogonal lifted from D12
ρ2122-2-20000-12-200001-11-1-13-33-3-33-3113    orthogonal lifted from D12
ρ222-22-20000-100-200211-1-33-13-3-1-331-331    orthogonal lifted from D12
ρ232-2-220000-10002-20-1113-3-3-1-1311-3-333    orthogonal lifted from D12
ρ242-2-220000-10002-20-111-333-1-1-31133-3-3    orthogonal lifted from D12
ρ252-2-220000-1000-220-111-33-3113-1-1-33-33    orthogonal lifted from D12
ρ262-22-20000-100-200211-13-3-1-33-13-313-31    orthogonal lifted from D12
ρ272-2-220000-1000-220-1113-3311-3-1-13-33-3    orthogonal lifted from D12
ρ2822-2-20000-1-2200001-1111-3-333-333-1-1-3    orthogonal lifted from D12
ρ292-22-20000-100200-211-1-331-3313-3-1-33-1    orthogonal lifted from D12
ρ3022-2-20000-12-200001-11-1-1-33-333-3311-3    orthogonal lifted from D12

Smallest permutation representation of C4⋊D12
On 48 points
Generators in S48
(1 24 37 27)(2 13 38 28)(3 14 39 29)(4 15 40 30)(5 16 41 31)(6 17 42 32)(7 18 43 33)(8 19 44 34)(9 20 45 35)(10 21 46 36)(11 22 47 25)(12 23 48 26)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 28)(14 27)(15 26)(16 25)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(37 39)(40 48)(41 47)(42 46)(43 45)

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

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

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

C4⋊D12 is a maximal subgroup of
C4.D24  C85D12  C4.5D24  C124D8  C8⋊D12  C42.19D6  Q85D12  C4⋊D24  Dic68D4  C127D8  Q82D12  C42.64D6  C42.70D6  C12⋊D8  C126SD16  C12.D8  C42.276D6  C4211D6  C42.100D6  D4×D12  Dic624D4  Q87D12  C42.136D6  C4220D6  C42.155D6  C4227D6  S3×C41D4  C42.240D6  C426D9  (C4×C12)⋊C6  C12⋊D12  C124D12  C20⋊D12  C426D15
C4⋊D12 is a maximal quotient of
(C2×C12)⋊5D4  (C2×C12).33D4  C85D12  C124D8  C8.8D12  C124Q16  C8⋊D12  C8.D12  C4210Dic3  (C2×C4)⋊6D12  C426D9  C12⋊D12  C124D12  C20⋊D12  C426D15

Matrix representation of C4⋊D12 in GL4(𝔽13) generated by

12000
01200
0036
00710
,
10300
10700
001212
0010
,
1100
01200
001212
0001
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,3,7,0,0,6,10],[10,10,0,0,3,7,0,0,0,0,12,1,0,0,12,0],[1,0,0,0,1,12,0,0,0,0,12,0,0,0,12,1] >;

C4⋊D12 in GAP, Magma, Sage, TeX

C_4\rtimes D_{12}
% in TeX

G:=Group("C4:D12");
// GroupNames label

G:=SmallGroup(96,81);
// by ID

G=gap.SmallGroup(96,81);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,50,2309]);
// Polycyclic

G:=Group<a,b,c|a^4=b^12=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Character table of C4⋊D12 in TeX

׿
×
𝔽