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G = C6213D4order 288 = 25·32

10th semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C6213D4, C62.255C23, (C6×D4)⋊14S3, (C2×C12)⋊16D6, (C22×C6)⋊7D6, C6.129(S3×D4), C34(C232D6), (C6×C12)⋊25C22, C3211C22≀C2, (C2×C62)⋊5C22, C625C418C2, C6.11D1225C2, C223(C327D4), (D4×C3×C6)⋊17C2, (C2×C3⋊S3)⋊17D4, C232(C2×C3⋊S3), C2.25(D4×C3⋊S3), (C2×D4)⋊3(C3⋊S3), (C2×C6)⋊8(C3⋊D4), (C23×C3⋊S3)⋊3C2, (C3×C6).283(C2×D4), C6.124(C2×C3⋊D4), (C2×C327D4)⋊10C2, (C2×C3⋊Dic3)⋊8C22, C2.13(C2×C327D4), (C2×C6).272(C22×S3), C22.59(C22×C3⋊S3), (C22×C3⋊S3).92C22, (C2×C4)⋊2(C2×C3⋊S3), SmallGroup(288,794)

Series: Derived Chief Lower central Upper central

C1C62 — C6213D4
C1C3C32C3×C6C62C22×C3⋊S3C23×C3⋊S3 — C6213D4
C32C62 — C6213D4
C1C22C2×D4

Generators and relations for C6213D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1772 in 390 conjugacy classes, 81 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×4], C4 [×3], C22, C22 [×2], C22 [×21], S3 [×16], C6 [×12], C6 [×12], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], C32, Dic3 [×8], C12 [×4], D6 [×64], C2×C6 [×12], C2×C6 [×20], C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×3], C2×Dic3 [×8], C3⋊D4 [×16], C2×C12 [×4], C3×D4 [×8], C22×S3 [×32], C22×C6 [×8], C22≀C2, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3 [×4], C2×C3⋊S3 [×12], C62, C62 [×2], C62 [×5], D6⋊C4 [×8], C6.D4 [×4], C2×C3⋊D4 [×8], C6×D4 [×4], S3×C23 [×4], C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12, D4×C32 [×2], C22×C3⋊S3 [×2], C22×C3⋊S3 [×6], C2×C62 [×2], C232D6 [×4], C6.11D12 [×2], C625C4, C2×C327D4 [×2], D4×C3×C6, C23×C3⋊S3, C6213D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×6], C23, D6 [×12], C2×D4 [×3], C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C22≀C2, C2×C3⋊S3 [×3], S3×D4 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C232D6 [×4], D4×C3⋊S3 [×2], C2×C327D4, C6213D4

Smallest permutation representation of C6213D4
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 39 61 21 15 58)(2 40 62 22 16 59)(3 41 63 23 17 60)(4 42 64 24 18 55)(5 37 65 19 13 56)(6 38 66 20 14 57)(7 31 45 67 26 49)(8 32 46 68 27 50)(9 33 47 69 28 51)(10 34 48 70 29 52)(11 35 43 71 30 53)(12 36 44 72 25 54)
(1 46 24 43)(2 49 19 52)(3 44 20 47)(4 53 21 50)(5 48 22 45)(6 51 23 54)(7 65 10 59)(8 55 11 61)(9 63 12 57)(13 29 40 26)(14 33 41 36)(15 27 42 30)(16 31 37 34)(17 25 38 28)(18 35 39 32)(56 70 62 67)(58 68 64 71)(60 72 66 69)
(2 6)(3 5)(7 25)(8 30)(9 29)(10 28)(11 27)(12 26)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(19 23)(20 22)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 46)(44 45)(47 48)(49 54)(50 53)(51 52)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C3D4A4B4C6A···6L6M···6AB12A···12H
order1222222222233334446···66···612···12
size1111224181818182222436362···24···44···4

54 irreducible representations

dim1111112222224
type++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6C3⋊D4S3×D4
kernelC6213D4C6.11D12C625C4C2×C327D4D4×C3×C6C23×C3⋊S3C6×D4C2×C3⋊S3C62C2×C12C22×C6C2×C6C6
# reps12121144248168

Matrix representation of C6213D4 in GL6(𝔽13)

0120000
110000
001100
0012000
000010
0000112
,
12120000
100000
001000
000100
0000120
0000012
,
1190000
1120000
0011900
0011200
0000122
0000121
,
1200000
110000
001000
00121200
000010
0000112

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[11,11,0,0,0,0,9,2,0,0,0,0,0,0,11,11,0,0,0,0,9,2,0,0,0,0,0,0,12,12,0,0,0,0,2,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12] >;

C6213D4 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{13}D_4
% in TeX

G:=Group("C6^2:13D4");
// GroupNames label

G:=SmallGroup(288,794);
// by ID

G=gap.SmallGroup(288,794);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,2693,9414]);
// 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^-1*b^3,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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