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G = C32⋊2C8⋊C4order 288 = 25·32

6th semidirect product of C32⋊2C8 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C32⋊2C8⋊C4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C32⋊2C8⋊C4
 Lower central C32 — C3×C6 — C32⋊2C8⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for C322C8⋊C4
G = < a,b,c,d | a3=b3=c8=d4=1, cbc-1=ab=ba, cac-1=a-1b, ad=da, bd=db, dcd-1=c5 >

Subgroups: 272 in 70 conjugacy classes, 24 normal (16 characteristic)
C1, C2 [×3], C3 [×2], C4 [×4], C22, C6 [×6], C8 [×4], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×4], C2×C6 [×2], C42, C2×C8 [×2], C3×C6 [×3], C2×Dic3 [×4], C2×C12 [×2], C8⋊C4, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, C62, C4×Dic3 [×2], C322C8 [×4], C2×C3⋊Dic3 [×2], C6×C12, C4×C3⋊Dic3, C2×C322C8 [×2], C322C8⋊C4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], C42, M4(2) [×2], C8⋊C4, C32⋊C4, C2×C32⋊C4, C32⋊M4(2), C4×C32⋊C4, C62.C4, C322C8⋊C4

Smallest permutation representation of C322C8⋊C4
On 96 points
Generators in S96
(2 44 55)(4 49 46)(6 48 51)(8 53 42)(10 85 34)(12 36 87)(14 81 38)(16 40 83)(18 75 96)(20 90 77)(22 79 92)(24 94 73)(25 64 66)(27 68 58)(29 60 70)(31 72 62)
(1 43 54)(2 44 55)(3 56 45)(4 49 46)(5 47 50)(6 48 51)(7 52 41)(8 53 42)(9 84 33)(10 85 34)(11 35 86)(12 36 87)(13 88 37)(14 81 38)(15 39 82)(16 40 83)(17 74 95)(18 75 96)(19 89 76)(20 90 77)(21 78 91)(22 79 92)(23 93 80)(24 94 73)(25 64 66)(26 67 57)(27 68 58)(28 59 69)(29 60 70)(30 71 61)(31 72 62)(32 63 65)
(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)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
(1 15 95 28)(2 12 96 25)(3 9 89 30)(4 14 90 27)(5 11 91 32)(6 16 92 29)(7 13 93 26)(8 10 94 31)(17 59 43 39)(18 64 44 36)(19 61 45 33)(20 58 46 38)(21 63 47 35)(22 60 48 40)(23 57 41 37)(24 62 42 34)(49 81 77 68)(50 86 78 65)(51 83 79 70)(52 88 80 67)(53 85 73 72)(54 82 74 69)(55 87 75 66)(56 84 76 71)

G:=sub<Sym(96)| (2,44,55)(4,49,46)(6,48,51)(8,53,42)(10,85,34)(12,36,87)(14,81,38)(16,40,83)(18,75,96)(20,90,77)(22,79,92)(24,94,73)(25,64,66)(27,68,58)(29,60,70)(31,72,62), (1,43,54)(2,44,55)(3,56,45)(4,49,46)(5,47,50)(6,48,51)(7,52,41)(8,53,42)(9,84,33)(10,85,34)(11,35,86)(12,36,87)(13,88,37)(14,81,38)(15,39,82)(16,40,83)(17,74,95)(18,75,96)(19,89,76)(20,90,77)(21,78,91)(22,79,92)(23,93,80)(24,94,73)(25,64,66)(26,67,57)(27,68,58)(28,59,69)(29,60,70)(30,71,61)(31,72,62)(32,63,65), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,15,95,28)(2,12,96,25)(3,9,89,30)(4,14,90,27)(5,11,91,32)(6,16,92,29)(7,13,93,26)(8,10,94,31)(17,59,43,39)(18,64,44,36)(19,61,45,33)(20,58,46,38)(21,63,47,35)(22,60,48,40)(23,57,41,37)(24,62,42,34)(49,81,77,68)(50,86,78,65)(51,83,79,70)(52,88,80,67)(53,85,73,72)(54,82,74,69)(55,87,75,66)(56,84,76,71)>;

G:=Group( (2,44,55)(4,49,46)(6,48,51)(8,53,42)(10,85,34)(12,36,87)(14,81,38)(16,40,83)(18,75,96)(20,90,77)(22,79,92)(24,94,73)(25,64,66)(27,68,58)(29,60,70)(31,72,62), (1,43,54)(2,44,55)(3,56,45)(4,49,46)(5,47,50)(6,48,51)(7,52,41)(8,53,42)(9,84,33)(10,85,34)(11,35,86)(12,36,87)(13,88,37)(14,81,38)(15,39,82)(16,40,83)(17,74,95)(18,75,96)(19,89,76)(20,90,77)(21,78,91)(22,79,92)(23,93,80)(24,94,73)(25,64,66)(26,67,57)(27,68,58)(28,59,69)(29,60,70)(30,71,61)(31,72,62)(32,63,65), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,15,95,28)(2,12,96,25)(3,9,89,30)(4,14,90,27)(5,11,91,32)(6,16,92,29)(7,13,93,26)(8,10,94,31)(17,59,43,39)(18,64,44,36)(19,61,45,33)(20,58,46,38)(21,63,47,35)(22,60,48,40)(23,57,41,37)(24,62,42,34)(49,81,77,68)(50,86,78,65)(51,83,79,70)(52,88,80,67)(53,85,73,72)(54,82,74,69)(55,87,75,66)(56,84,76,71) );

G=PermutationGroup([(2,44,55),(4,49,46),(6,48,51),(8,53,42),(10,85,34),(12,36,87),(14,81,38),(16,40,83),(18,75,96),(20,90,77),(22,79,92),(24,94,73),(25,64,66),(27,68,58),(29,60,70),(31,72,62)], [(1,43,54),(2,44,55),(3,56,45),(4,49,46),(5,47,50),(6,48,51),(7,52,41),(8,53,42),(9,84,33),(10,85,34),(11,35,86),(12,36,87),(13,88,37),(14,81,38),(15,39,82),(16,40,83),(17,74,95),(18,75,96),(19,89,76),(20,90,77),(21,78,91),(22,79,92),(23,93,80),(24,94,73),(25,64,66),(26,67,57),(27,68,58),(28,59,69),(29,60,70),(30,71,61),(31,72,62),(32,63,65)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,15,95,28),(2,12,96,25),(3,9,89,30),(4,14,90,27),(5,11,91,32),(6,16,92,29),(7,13,93,26),(8,10,94,31),(17,59,43,39),(18,64,44,36),(19,61,45,33),(20,58,46,38),(21,63,47,35),(22,60,48,40),(23,57,41,37),(24,62,42,34),(49,81,77,68),(50,86,78,65),(51,83,79,70),(52,88,80,67),(53,85,73,72),(54,82,74,69),(55,87,75,66),(56,84,76,71)])

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 8A ··· 8H 12A ··· 12H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 4 4 2 2 9 9 9 9 18 18 4 ··· 4 18 ··· 18 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 4 4 4 type + + + + + - image C1 C2 C2 C4 C4 C4 M4(2) C32⋊C4 C2×C32⋊C4 C32⋊M4(2) C4×C32⋊C4 C62.C4 kernel C32⋊2C8⋊C4 C4×C3⋊Dic3 C2×C32⋊2C8 C32⋊2C8 C2×C3⋊Dic3 C6×C12 C3×C6 C2×C4 C22 C2 C2 C2 # reps 1 1 2 8 2 2 4 2 2 4 4 4

Matrix representation of C322C8⋊C4 in GL6(𝔽73)

 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 0 1 0 0 46 46 72 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 27 0 1 0 0 46 0 72 72
,
 22 0 0 0 0 0 10 51 0 0 0 0 0 0 0 0 1 72 0 0 27 27 2 1 0 0 35 43 46 0 0 0 43 69 46 0
,
 27 71 0 0 0 0 72 46 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 25 25 46 0 0 0 25 25 0 46

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,46,0,0,0,1,0,46,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,46,0,0,72,72,27,0,0,0,0,0,0,72,0,0,0,0,1,72],[22,10,0,0,0,0,0,51,0,0,0,0,0,0,0,27,35,43,0,0,0,27,43,69,0,0,1,2,46,46,0,0,72,1,0,0],[27,72,0,0,0,0,71,46,0,0,0,0,0,0,27,0,25,25,0,0,0,27,25,25,0,0,0,0,46,0,0,0,0,0,0,46] >;

C322C8⋊C4 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2C_8\rtimes C_4
% in TeX

G:=Group("C3^2:2C8:C4");
// GroupNames label

G:=SmallGroup(288,425);
// by ID

G=gap.SmallGroup(288,425);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,100,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^4=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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