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G = C6212D4order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C6212D4, C62.226C23, (C2×C12)⋊2D6, (C2×C6)⋊6D12, (C6×C12)⋊2C22, C6.52(C2×D12), C6.108(S3×D4), C329C22≀C2, C32(D6⋊D4), (C22×C6).88D6, C6.11D124C2, C223(C12⋊S3), (C2×C62).65C22, C2.7(D4×C3⋊S3), (C2×C3⋊S3)⋊16D4, (C23×C3⋊S3)⋊2C2, (C3×C22⋊C4)⋊3S3, (C2×C12⋊S3)⋊4C2, C22⋊C42(C3⋊S3), C2.7(C2×C12⋊S3), (C3×C6).192(C2×D4), C23.20(C2×C3⋊S3), (C2×C327D4)⋊7C2, (C32×C22⋊C4)⋊4C2, (C22×C3⋊S3)⋊3C22, (C2×C3⋊Dic3)⋊7C22, (C2×C6).243(C22×S3), C22.41(C22×C3⋊S3), (C2×C4)⋊1(C2×C3⋊S3), SmallGroup(288,739)

Series: Derived Chief Lower central Upper central

C1C62 — C6212D4
C1C3C32C3×C6C62C22×C3⋊S3C23×C3⋊S3 — C6212D4
C32C62 — C6212D4
C1C22C22⋊C4

Generators and relations for C6212D4
 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 [×2], C2 [×7], C3 [×4], C4 [×3], C22, C22 [×2], C22 [×21], S3 [×20], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], C32, Dic3 [×4], C12 [×8], D6 [×76], C2×C6 [×12], C2×C6 [×8], C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, C3⋊S3 [×5], C3×C6, C3×C6 [×2], C3×C6 [×2], D12 [×16], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×8], C22×S3 [×36], C22×C6 [×4], C22≀C2, C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×15], C62, C62 [×2], C62 [×2], D6⋊C4 [×8], C3×C22⋊C4 [×4], C2×D12 [×8], C2×C3⋊D4 [×4], S3×C23 [×4], C12⋊S3 [×4], C2×C3⋊Dic3, C327D4 [×2], C6×C12 [×2], C22×C3⋊S3, C22×C3⋊S3 [×2], C22×C3⋊S3 [×6], C2×C62, D6⋊D4 [×4], C6.11D12 [×2], C32×C22⋊C4, C2×C12⋊S3 [×2], C2×C327D4, C23×C3⋊S3, C6212D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×6], C23, D6 [×12], C2×D4 [×3], C3⋊S3, D12 [×8], C22×S3 [×4], C22≀C2, C2×C3⋊S3 [×3], C2×D12 [×4], S3×D4 [×8], C12⋊S3 [×2], C22×C3⋊S3, D6⋊D4 [×4], C2×C12⋊S3, D4×C3⋊S3 [×2], C6212D4

Smallest permutation representation of C6212D4
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 26 29 33 35 20)(2 27 30 31 36 21)(3 25 28 32 34 19)(4 10 8 16 14 23)(5 11 9 17 15 24)(6 12 7 18 13 22)(37 44 50 40 47 53)(38 45 51 41 48 54)(39 46 52 42 43 49)(55 69 65 58 72 62)(56 70 66 59 67 63)(57 71 61 60 68 64)
(1 38 24 66)(2 42 22 64)(3 40 23 62)(4 55 25 47)(5 59 26 45)(6 57 27 43)(7 61 31 39)(8 65 32 37)(9 63 33 41)(10 69 28 53)(11 67 29 51)(12 71 30 49)(13 68 21 52)(14 72 19 50)(15 70 20 54)(16 58 34 44)(17 56 35 48)(18 60 36 46)
(1 61)(2 63)(3 65)(4 53)(5 49)(6 51)(7 38)(8 40)(9 42)(10 47)(11 43)(12 45)(13 48)(14 44)(15 46)(16 50)(17 52)(18 54)(19 58)(20 60)(21 56)(22 41)(23 37)(24 39)(25 69)(26 71)(27 67)(28 55)(29 57)(30 59)(31 66)(32 62)(33 64)(34 72)(35 68)(36 70)

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim1111112222224
type+++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D12S3×D4
kernelC6212D4C6.11D12C32×C22⋊C4C2×C12⋊S3C2×C327D4C23×C3⋊S3C3×C22⋊C4C2×C3⋊S3C62C2×C12C22×C6C2×C6C6
# reps12121144284168

Matrix representation of C6212D4 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] >;

C6212D4 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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