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G = C122⋊C2order 288 = 25·32

7th semidirect product of C122 and C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: C1227C2, C12.58D12, C62.109D4, (C4×C12)⋊8S3, C329C4≀C2, C12⋊S34C4, C12.60(C4×S3), C424(C3⋊S3), C324Q84C4, C6.20(D6⋊C4), (C3×C12).138D4, (C2×C12).396D6, C32(C424S3), C12.58D61C2, C4.17(C12⋊S3), (C6×C12).306C22, C12.59D6.1C2, C2.3(C6.11D12), C22.7(C327D4), C4.6(C4×C3⋊S3), (C3×C12).91(C2×C4), (C2×C6).85(C3⋊D4), (C3×C6).51(C22⋊C4), (C2×C4).70(C2×C3⋊S3), SmallGroup(288,280)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C122⋊C2
C1C3C32C3×C6C62C6×C12C12.59D6 — C122⋊C2
C32C3×C6C3×C12 — C122⋊C2
C1C4C2×C4C42

Generators and relations for C122⋊C2
 G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5b9, cbc=b-1 >

Subgroups: 508 in 132 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, S3 [×4], C6 [×4], C6 [×4], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C32, Dic3 [×4], C12 [×8], C12 [×8], D6 [×4], C2×C6 [×4], C42, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×4], Dic6 [×4], C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×4], C4≀C2, C3⋊Dic3, C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3, C62, C4.Dic3 [×4], C4×C12 [×4], C4○D12 [×4], C324C8, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C6×C12, C424S3 [×4], C12.58D6, C122, C12.59D6, C122⋊C2
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C4≀C2, C2×C3⋊S3, D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C424S3 [×4], C6.11D12, C122⋊C2

Smallest permutation representation of C122⋊C2
On 72 points
Generators in S72
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 33 12 27 15 21 4 36 9 30 18 24)(2 34 7 28 16 22 5 31 10 25 13 19)(3 35 8 29 17 23 6 32 11 26 14 20)(37 53 64 40 56 67 43 59 70 46 50 61)(38 54 65 41 57 68 44 60 71 47 51 62)(39 55 66 42 58 69 45 49 72 48 52 63)
(1 55)(2 57)(3 59)(4 49)(5 51)(6 53)(7 65)(8 67)(9 69)(10 71)(11 61)(12 63)(13 44)(14 46)(15 48)(16 38)(17 40)(18 42)(19 68)(20 70)(21 72)(22 62)(23 64)(24 66)(25 60)(26 50)(27 52)(28 54)(29 56)(30 58)(31 47)(32 37)(33 39)(34 41)(35 43)(36 45)

G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,33,12,27,15,21,4,36,9,30,18,24)(2,34,7,28,16,22,5,31,10,25,13,19)(3,35,8,29,17,23,6,32,11,26,14,20)(37,53,64,40,56,67,43,59,70,46,50,61)(38,54,65,41,57,68,44,60,71,47,51,62)(39,55,66,42,58,69,45,49,72,48,52,63), (1,55)(2,57)(3,59)(4,49)(5,51)(6,53)(7,65)(8,67)(9,69)(10,71)(11,61)(12,63)(13,44)(14,46)(15,48)(16,38)(17,40)(18,42)(19,68)(20,70)(21,72)(22,62)(23,64)(24,66)(25,60)(26,50)(27,52)(28,54)(29,56)(30,58)(31,47)(32,37)(33,39)(34,41)(35,43)(36,45)>;

G:=Group( (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,33,12,27,15,21,4,36,9,30,18,24)(2,34,7,28,16,22,5,31,10,25,13,19)(3,35,8,29,17,23,6,32,11,26,14,20)(37,53,64,40,56,67,43,59,70,46,50,61)(38,54,65,41,57,68,44,60,71,47,51,62)(39,55,66,42,58,69,45,49,72,48,52,63), (1,55)(2,57)(3,59)(4,49)(5,51)(6,53)(7,65)(8,67)(9,69)(10,71)(11,61)(12,63)(13,44)(14,46)(15,48)(16,38)(17,40)(18,42)(19,68)(20,70)(21,72)(22,62)(23,64)(24,66)(25,60)(26,50)(27,52)(28,54)(29,56)(30,58)(31,47)(32,37)(33,39)(34,41)(35,43)(36,45) );

G=PermutationGroup([(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,33,12,27,15,21,4,36,9,30,18,24),(2,34,7,28,16,22,5,31,10,25,13,19),(3,35,8,29,17,23,6,32,11,26,14,20),(37,53,64,40,56,67,43,59,70,46,50,61),(38,54,65,41,57,68,44,60,71,47,51,62),(39,55,66,42,58,69,45,49,72,48,52,63)], [(1,55),(2,57),(3,59),(4,49),(5,51),(6,53),(7,65),(8,67),(9,69),(10,71),(11,61),(12,63),(13,44),(14,46),(15,48),(16,38),(17,40),(18,42),(19,68),(20,70),(21,72),(22,62),(23,64),(24,66),(25,60),(26,50),(27,52),(28,54),(29,56),(30,58),(31,47),(32,37),(33,39),(34,41),(35,43),(36,45)])

78 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C···4G4H6A···6L8A8B12A···12AV
order12223333444···446···68812···12
size112362222112···2362···236362···2

78 irreducible representations

dim111111222222222
type+++++++++
imageC1C2C2C2C4C4S3D4D4D6C4×S3D12C3⋊D4C4≀C2C424S3
kernelC122⋊C2C12.58D6C122C12.59D6C324Q8C12⋊S3C4×C12C3×C12C62C2×C12C12C12C2×C6C32C3
# reps1111224114888432

Matrix representation of C122⋊C2 in GL4(𝔽73) generated by

9000
07000
0010
00072
,
46000
02700
00650
0009
,
0100
1000
0001
0010
G:=sub<GL(4,GF(73))| [9,0,0,0,0,70,0,0,0,0,1,0,0,0,0,72],[46,0,0,0,0,27,0,0,0,0,65,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C122⋊C2 in GAP, Magma, Sage, TeX

C_{12}^2\rtimes C_2
% in TeX

G:=Group("C12^2:C2");
// GroupNames label

G:=SmallGroup(288,280);
// by ID

G=gap.SmallGroup(288,280);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^5*b^9,c*b*c=b^-1>;
// generators/relations

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