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G = C22×C32⋊2C8order 288 = 25·32

Direct product of C22 and C32⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C32⋊2C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8 — C22×C32⋊2C8
 Lower central C32 — C22×C32⋊2C8
 Upper central C1 — C23

Generators and relations for C22×C322C8
G = < a,b,c,d,e | a2=b2=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 448 in 130 conjugacy classes, 54 normal (9 characteristic)
C1, C2, C2 [×6], C3 [×2], C4 [×4], C22 [×7], C6 [×14], C8 [×4], C2×C4 [×6], C23, C32, Dic3 [×8], C2×C6 [×14], C2×C8 [×6], C22×C4, C3×C6, C3×C6 [×6], C2×Dic3 [×12], C22×C6 [×2], C22×C8, C3⋊Dic3, C3⋊Dic3 [×3], C62 [×7], C22×Dic3 [×2], C322C8 [×4], C2×C3⋊Dic3 [×6], C2×C62, C2×C322C8 [×6], C22×C3⋊Dic3, C22×C322C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, C22×C8, C32⋊C4, C322C8 [×4], C2×C32⋊C4 [×3], C2×C322C8 [×6], C22×C32⋊C4, C22×C322C8

Smallest permutation representation of C22×C322C8
On 96 points
Generators in S96
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 89)(88 90)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(33 71)(34 72)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 89)(64 90)
(2 80 30)(4 32 74)(6 76 26)(8 28 78)(10 84 50)(12 52 86)(14 88 54)(16 56 82)(18 34 92)(20 94 36)(22 38 96)(24 90 40)(42 64 70)(44 72 58)(46 60 66)(48 68 62)
(1 79 29)(2 80 30)(3 31 73)(4 32 74)(5 75 25)(6 76 26)(7 27 77)(8 28 78)(9 83 49)(10 84 50)(11 51 85)(12 52 86)(13 87 53)(14 88 54)(15 55 81)(16 56 82)(17 33 91)(18 34 92)(19 93 35)(20 94 36)(21 37 95)(22 38 96)(23 89 39)(24 90 40)(41 63 69)(42 64 70)(43 71 57)(44 72 58)(45 59 65)(46 60 66)(47 67 61)(48 68 62)
(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)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 6A ··· 6N 8A ··· 8P order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 8 ··· 8 size 1 1 ··· 1 4 4 9 ··· 9 4 ··· 4 9 ··· 9

48 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C8 C32⋊C4 C32⋊2C8 C2×C32⋊C4 kernel C22×C32⋊2C8 C2×C32⋊2C8 C22×C3⋊Dic3 C2×C3⋊Dic3 C2×C62 C62 C23 C22 C22 # reps 1 6 1 6 2 16 2 8 6

Matrix representation of C22×C322C8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 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
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 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
,
 63 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 1 0 0 46 46 71 72 0 0 32 19 27 0 0 0 19 21 27 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[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],[63,0,0,0,0,0,0,72,0,0,0,0,0,0,0,46,32,19,0,0,0,46,19,21,0,0,72,71,27,27,0,0,1,72,0,0] >;

C22×C322C8 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("C2^2xC3^2:2C8");
// GroupNames label

G:=SmallGroup(288,939);
// by ID

G=gap.SmallGroup(288,939);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,80,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
// generators/relations

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