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G = C62:12D4order 288 = 25·32

9th semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62:12D4, C62.226C23, (C2xC12):2D6, (C2xC6):6D12, (C6xC12):2C22, C6.52(C2xD12), C6.108(S3xD4), C32:9C22wrC2, C3:2(D6:D4), (C22xC6).88D6, C6.11D12:4C2, C22:3(C12:S3), (C2xC62).65C22, C2.7(D4xC3:S3), (C2xC3:S3):16D4, (C23xC3:S3):2C2, (C3xC22:C4):3S3, (C2xC12:S3):4C2, C22:C4:2(C3:S3), C2.7(C2xC12:S3), (C3xC6).192(C2xD4), C23.20(C2xC3:S3), (C2xC32:7D4):7C2, (C32xC22:C4):4C2, (C22xC3:S3):3C22, (C2xC3:Dic3):7C22, (C2xC6).243(C22xS3), C22.41(C22xC3:S3), (C2xC4):1(C2xC3:S3), SmallGroup(288,739)

Series: Derived Chief Lower central Upper central

C1C62 — C62:12D4
C1C3C32C3xC6C62C22xC3:S3C23xC3:S3 — C62:12D4
C32C62 — C62:12D4
C1C22C22:C4

Generators and relations for C62:12D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=ab3, dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1964 in 390 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C22:C4, C2xD4, C24, C3:S3, C3xC6, C3xC6, C3xC6, D12, C2xDic3, C3:D4, C2xC12, C22xS3, C22xC6, C22wrC2, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, C62, C62, D6:C4, C3xC22:C4, C2xD12, C2xC3:D4, S3xC23, C12:S3, C2xC3:Dic3, C32:7D4, C6xC12, C22xC3:S3, C22xC3:S3, C22xC3:S3, C2xC62, D6:D4, C6.11D12, C32xC22:C4, C2xC12:S3, C2xC32:7D4, C23xC3:S3, C62:12D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, D12, C22xS3, C22wrC2, C2xC3:S3, C2xD12, S3xD4, C12:S3, C22xC3:S3, D6:D4, C2xC12:S3, D4xC3:S3, C62:12D4

Smallest permutation representation of C62:12D4
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 17 10 23 13 19)(2 18 11 24 14 20)(3 16 12 22 15 21)(4 33 25 29 8 36)(5 31 26 30 9 34)(6 32 27 28 7 35)(37 46 69 40 43 72)(38 47 70 41 44 67)(39 48 71 42 45 68)(49 65 59 52 62 56)(50 66 60 53 63 57)(51 61 55 54 64 58)
(1 70 9 54)(2 68 7 52)(3 72 8 50)(4 60 12 46)(5 58 10 44)(6 56 11 48)(13 38 26 61)(14 42 27 65)(15 40 25 63)(16 37 36 66)(17 41 34 64)(18 39 35 62)(19 47 30 55)(20 45 28 59)(21 43 29 57)(22 69 33 53)(23 67 31 51)(24 71 32 49)
(1 51)(2 53)(3 49)(4 39)(5 41)(6 37)(7 69)(8 71)(9 67)(10 64)(11 66)(12 62)(13 55)(14 57)(15 59)(16 56)(17 58)(18 60)(19 61)(20 63)(21 65)(22 52)(23 54)(24 50)(25 45)(26 47)(27 43)(28 40)(29 42)(30 38)(31 70)(32 72)(33 68)(34 44)(35 46)(36 48)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C3D4A4B4C6A···6L6M···6T12A···12P
order1222222222233334446···66···612···12
size1111221818181836222244362···24···44···4

54 irreducible representations

dim1111112222224
type+++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D12S3xD4
kernelC62:12D4C6.11D12C32xC22:C4C2xC12:S3C2xC32:7D4C23xC3:S3C3xC22:C4C2xC3:S3C62C2xC12C22xC6C2xC6C6
# reps12121144284168

Matrix representation of C62:12D4 in GL6(F13)

12120000
100000
001000
000100
000010
00001212
,
12120000
100000
00121200
001000
0000120
0000012
,
360000
7100000
003600
0071000
00001211
000011
,
360000
3100000
0031000
0071000
000012
0000012

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,1,0,0,0,0,0,2,12] >;

C62:12D4 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{12}D_4
% in TeX

G:=Group("C6^2:12D4");
// GroupNames label

G:=SmallGroup(288,739);
// by ID

G=gap.SmallGroup(288,739);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^3,d*a*d=a^-1*b^3,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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