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G = C22×C3⋊D12order 288 = 25·32

Direct product of C22 and C3⋊D12

direct product, metabelian, supersoluble, monomial

Aliases: C22×C3⋊D12, C6216D4, C62.146C23, C63(C2×D12), (C2×C6)⋊12D12, C23.47S32, (S3×C23)⋊7S3, (S3×C6)⋊7C23, D66(C22×S3), C33(C22×D12), C326(C22×D4), (C2×Dic3)⋊22D6, (C22×S3)⋊15D6, (C3×C6).33C24, C6.33(S3×C23), Dic35(C22×S3), (C3×Dic3)⋊5C23, (C22×C6).123D6, (C6×Dic3)⋊27C22, (C22×Dic3)⋊12S3, (C2×C62).81C22, (C3×C6)⋊5(C2×D4), C61(C2×C3⋊D4), (S3×C22×C6)⋊6C2, (C23×C3⋊S3)⋊4C2, (C2×C3⋊S3)⋊4C23, (S3×C2×C6)⋊17C22, C22.70(C2×S32), C2.33(C22×S32), (Dic3×C2×C6)⋊12C2, C31(C22×C3⋊D4), (C2×C6)⋊12(C3⋊D4), (C22×C3⋊S3)⋊13C22, (C2×C6).161(C22×S3), SmallGroup(288,974)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C22×C3⋊D12
C1C3C32C3×C6S3×C6C3⋊D12C2×C3⋊D12 — C22×C3⋊D12
C32C3×C6 — C22×C3⋊D12
C1C23

Generators and relations for C22×C3⋊D12
 G = < a,b,c,d,e | a2=b2=c3=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 2178 in 539 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×2], C3, C4 [×4], C22 [×7], C22 [×32], S3 [×20], C6 [×2], C6 [×12], C6 [×11], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, Dic3 [×4], C12 [×4], D6 [×4], D6 [×72], C2×C6 [×14], C2×C6 [×23], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3⋊S3 [×4], C3×C6, C3×C6 [×6], D12 [×16], C2×Dic3 [×6], C3⋊D4 [×16], C2×C12 [×6], C22×S3 [×6], C22×S3 [×38], C22×C6 [×2], C22×C6 [×11], C22×D4, C3×Dic3 [×4], S3×C6 [×4], S3×C6 [×12], C2×C3⋊S3 [×4], C2×C3⋊S3 [×12], C62 [×7], C2×D12 [×12], C22×Dic3, C2×C3⋊D4 [×12], C22×C12, S3×C23, S3×C23 [×3], C23×C6, C3⋊D12 [×16], C6×Dic3 [×6], S3×C2×C6 [×6], S3×C2×C6 [×4], C22×C3⋊S3 [×6], C22×C3⋊S3 [×4], C2×C62, C22×D12, C22×C3⋊D4, C2×C3⋊D12 [×12], Dic3×C2×C6, S3×C22×C6, C23×C3⋊S3, C22×C3⋊D12
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, D12 [×4], C3⋊D4 [×4], C22×S3 [×14], C22×D4, S32, C2×D12 [×6], C2×C3⋊D4 [×6], S3×C23 [×2], C3⋊D12 [×4], C2×S32 [×3], C22×D12, C22×C3⋊D4, C2×C3⋊D12 [×6], C22×S32, C22×C3⋊D12

Smallest permutation representation of C22×C3⋊D12
On 48 points
Generators in S48
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 21 17)(14 18 22)(15 23 19)(16 20 24)(25 29 33)(26 34 30)(27 31 35)(28 36 32)(37 41 45)(38 46 42)(39 43 47)(40 48 44)
(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 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)

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

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

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

60 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O3A3B3C4A4B4C4D6A···6N6O···6U6V···6AC12A···12H
order12···22222222233344446···66···66···612···12
size11···166661818181822466662···24···46···66···6

60 irreducible representations

dim1111122222222444
type+++++++++++++++
imageC1C2C2C2C2S3S3D4D6D6D6D12C3⋊D4S32C3⋊D12C2×S32
kernelC22×C3⋊D12C2×C3⋊D12Dic3×C2×C6S3×C22×C6C23×C3⋊S3C22×Dic3S3×C23C62C2×Dic3C22×S3C22×C6C2×C6C2×C6C23C22C22
# reps11211111466288143

Matrix representation of C22×C3⋊D12 in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
001000
000100
0000-1-1
000010
,
1-10000
100000
000100
00-1000
000010
0000-1-1
,
-100000
-110000
001000
000-100
0000-10
000011

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1] >;

C22×C3⋊D12 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes D_{12}
% in TeX

G:=Group("C2^2xC3:D12");
// GroupNames label

G:=SmallGroup(288,974);
// by ID

G=gap.SmallGroup(288,974);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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