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## G = C22⋊C4×C3⋊S3order 288 = 25·32

### Direct product of C22⋊C4 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C22⋊C4×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C23×C3⋊S3 — C22⋊C4×C3⋊S3
 Lower central C32 — C3×C6 — C22⋊C4×C3⋊S3
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22⋊C4×C3⋊S3
G = < a,b,c,d,e,f | a2=b2=c4=d3=e3=f2=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1700 in 396 conjugacy classes, 93 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C3 [×4], C4 [×4], C22, C22 [×2], C22 [×20], S3 [×24], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C32, Dic3 [×8], C12 [×8], D6 [×72], C2×C6 [×12], C2×C6 [×8], C22⋊C4, C22⋊C4 [×3], C22×C4 [×2], C24, C3⋊S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×16], C2×Dic3 [×8], C2×C12 [×8], C22×S3 [×40], C22×C6 [×4], C2×C22⋊C4, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×8], C2×C3⋊S3 [×10], C62, C62 [×2], C62 [×2], D6⋊C4 [×8], C6.D4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×8], S3×C23 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3 [×2], C6×C12 [×2], C22×C3⋊S3 [×2], C22×C3⋊S3 [×4], C22×C3⋊S3 [×4], C2×C62, S3×C22⋊C4 [×4], C6.11D12 [×2], C625C4, C32×C22⋊C4, C2×C4×C3⋊S3 [×2], C23×C3⋊S3, C22⋊C4×C3⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×4], C23, D6 [×12], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C3⋊S3, C4×S3 [×8], C22×S3 [×4], C2×C22⋊C4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], S3×D4 [×8], C4×C3⋊S3 [×2], C22×C3⋊S3, S3×C22⋊C4 [×4], C2×C4×C3⋊S3, D4×C3⋊S3 [×2], C22⋊C4×C3⋊S3

Smallest permutation representation of C22⋊C4×C3⋊S3
On 72 points
Generators in S72
(1 3)(2 17)(4 19)(5 7)(6 66)(8 68)(9 11)(10 39)(12 37)(13 15)(14 46)(16 48)(18 20)(21 23)(22 34)(24 36)(25 27)(26 55)(28 53)(29 31)(30 42)(32 44)(33 35)(38 40)(41 43)(45 47)(49 51)(50 64)(52 62)(54 56)(57 59)(58 70)(60 72)(61 63)(65 67)(69 71)
(1 18)(2 19)(3 20)(4 17)(5 67)(6 68)(7 65)(8 66)(9 40)(10 37)(11 38)(12 39)(13 47)(14 48)(15 45)(16 46)(21 35)(22 36)(23 33)(24 34)(25 56)(26 53)(27 54)(28 55)(29 43)(30 44)(31 41)(32 42)(49 61)(50 62)(51 63)(52 64)(57 71)(58 72)(59 69)(60 70)
(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 43 35)(2 44 36)(3 41 33)(4 42 34)(5 25 13)(6 26 14)(7 27 15)(8 28 16)(9 51 71)(10 52 72)(11 49 69)(12 50 70)(17 32 24)(18 29 21)(19 30 22)(20 31 23)(37 64 58)(38 61 59)(39 62 60)(40 63 57)(45 65 54)(46 66 55)(47 67 56)(48 68 53)
(1 51 7)(2 52 8)(3 49 5)(4 50 6)(9 15 35)(10 16 36)(11 13 33)(12 14 34)(17 62 68)(18 63 65)(19 64 66)(20 61 67)(21 40 45)(22 37 46)(23 38 47)(24 39 48)(25 41 69)(26 42 70)(27 43 71)(28 44 72)(29 57 54)(30 58 55)(31 59 56)(32 60 53)
(5 49)(6 50)(7 51)(8 52)(9 27)(10 28)(11 25)(12 26)(13 69)(14 70)(15 71)(16 72)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 55)(38 56)(39 53)(40 54)(45 57)(46 58)(47 59)(48 60)(61 67)(62 68)(63 65)(64 66)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 6M ··· 6T 12A ··· 12P order 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 9 9 9 9 18 18 2 2 2 2 2 2 2 2 18 18 18 18 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4×S3 S3×D4 kernel C22⋊C4×C3⋊S3 C6.11D12 C62⋊5C4 C32×C22⋊C4 C2×C4×C3⋊S3 C23×C3⋊S3 C22×C3⋊S3 C3×C22⋊C4 C2×C3⋊S3 C2×C12 C22×C6 C2×C6 C6 # reps 1 2 1 1 2 1 8 4 4 8 4 16 8

Matrix representation of C22⋊C4×C3⋊S3 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 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
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22⋊C4×C3⋊S3 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_3\rtimes S_3
% in TeX

G:=Group("C2^2:C4xC3:S3");
// GroupNames label

G:=SmallGroup(288,737);
// by ID

G=gap.SmallGroup(288,737);
# by ID

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

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

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