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G = C6224D4order 288 = 25·32

3rd semidirect product of C62 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial

Aliases: C6224D4, C62.264C23, (C23×C6)⋊9S3, C245(C3⋊S3), C3213C22≀C2, (C22×C62)⋊4C2, C625C421C2, C33(C244S3), (C22×C6).162D6, C224(C327D4), (C2×C62).114C22, (C2×C6)⋊17(C3⋊D4), (C3×C6).297(C2×D4), C6.138(C2×C3⋊D4), C23.31(C2×C3⋊S3), (C2×C327D4)⋊14C2, (C22×C3⋊S3)⋊4C22, (C2×C3⋊Dic3)⋊9C22, C2.26(C2×C327D4), (C2×C6).281(C22×S3), C22.66(C22×C3⋊S3), SmallGroup(288,810)

Series: Derived Chief Lower central Upper central

C1C62 — C6224D4
C1C3C32C3×C6C62C22×C3⋊S3C2×C327D4 — C6224D4
C32C62 — C6224D4
C1C22C24

Generators and relations for C6224D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1260 in 390 conjugacy classes, 101 normal (8 characteristic)
C1, C2 [×3], C2 [×7], C3 [×4], C4 [×3], C22, C22 [×6], C22 [×17], S3 [×4], C6 [×12], C6 [×24], C2×C4 [×3], D4 [×6], C23 [×3], C23 [×7], C32, Dic3 [×12], D6 [×12], C2×C6 [×28], C2×C6 [×56], C22⋊C4 [×3], C2×D4 [×3], C24, C3⋊S3, C3×C6 [×3], C3×C6 [×6], C2×Dic3 [×12], C3⋊D4 [×24], C22×S3 [×4], C22×C6 [×12], C22×C6 [×24], C22≀C2, C3⋊Dic3 [×3], C2×C3⋊S3 [×3], C62, C62 [×6], C62 [×14], C6.D4 [×12], C2×C3⋊D4 [×12], C23×C6 [×4], C2×C3⋊Dic3 [×3], C327D4 [×6], C22×C3⋊S3, C2×C62 [×3], C2×C62 [×6], C244S3 [×4], C625C4 [×3], C2×C327D4 [×3], C22×C62, C6224D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×6], C23, D6 [×12], C2×D4 [×3], C3⋊S3, C3⋊D4 [×24], C22×S3 [×4], C22≀C2, C2×C3⋊S3 [×3], C2×C3⋊D4 [×12], C327D4 [×6], C22×C3⋊S3, C244S3 [×4], C2×C327D4 [×3], C6224D4

Smallest permutation representation of C6224D4
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 18 59 54 33 46)(2 13 60 49 34 47)(3 14 55 50 35 48)(4 15 56 51 36 43)(5 16 57 52 31 44)(6 17 58 53 32 45)(7 64 27 22 71 39)(8 65 28 23 72 40)(9 66 29 24 67 41)(10 61 30 19 68 42)(11 62 25 20 69 37)(12 63 26 21 70 38)
(1 41 4 26)(2 28 5 37)(3 39 6 30)(7 45 19 48)(8 57 20 60)(9 43 21 46)(10 55 22 58)(11 47 23 44)(12 59 24 56)(13 65 16 69)(14 71 17 61)(15 63 18 67)(25 49 40 52)(27 53 42 50)(29 51 38 54)(31 62 34 72)(32 68 35 64)(33 66 36 70)
(1 38)(2 25)(3 42)(4 29)(5 40)(6 27)(7 58)(8 44)(9 56)(10 48)(11 60)(12 46)(13 62)(14 68)(15 66)(16 72)(17 64)(18 70)(19 55)(20 47)(21 59)(22 45)(23 57)(24 43)(26 54)(28 52)(30 50)(31 65)(32 71)(33 63)(34 69)(35 61)(36 67)(37 49)(39 53)(41 51)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D···2I2J3A3B3C3D4A4B4C6A···6BH
order12222···2233334446···6
size11112···23622223636362···2

78 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2S3D4D6C3⋊D4
kernelC6224D4C625C4C2×C327D4C22×C62C23×C6C62C22×C6C2×C6
# reps1331461248

Matrix representation of C6224D4 in GL4(𝔽13) generated by

4000
0300
0030
0094
,
10000
0400
00120
00012
,
01200
1000
0065
0067
,
0100
1000
0078
0076
G:=sub<GL(4,GF(13))| [4,0,0,0,0,3,0,0,0,0,3,9,0,0,0,4],[10,0,0,0,0,4,0,0,0,0,12,0,0,0,0,12],[0,1,0,0,12,0,0,0,0,0,6,6,0,0,5,7],[0,1,0,0,1,0,0,0,0,0,7,7,0,0,8,6] >;

C6224D4 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{24}D_4
% in TeX

G:=Group("C6^2:24D4");
// GroupNames label

G:=SmallGroup(288,810);
// by ID

G=gap.SmallGroup(288,810);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,2693,9414]);
// Polycyclic

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

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