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## G = C2×C32⋊2D8order 288 = 25·32

### Direct product of C2 and C32⋊2D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊2D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C32⋊2D8 — C2×C32⋊2D8
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊2D8
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C322D8
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 626 in 163 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C8 [×2], C2×C4, D4 [×6], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×8], C2×C6 [×2], C2×C6 [×9], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×4], C3×C6, C3×C6 [×2], C3⋊C8 [×6], D12 [×4], D12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×6], C22×S3 [×2], C22×C6 [×2], C2×D8, C3×C12 [×2], S3×C6 [×8], C62, C2×C3⋊C8 [×3], D4⋊S3 [×8], C2×D12 [×2], C6×D4 [×2], C324C8 [×2], C3×D12 [×4], C3×D12 [×2], C6×C12, S3×C2×C6 [×2], C2×D4⋊S3 [×2], C322D8 [×4], C2×C324C8, C6×D12 [×2], C2×C322D8
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], D8 [×2], C2×D4, C3⋊D4 [×4], C22×S3 [×2], C2×D8, S32, D4⋊S3 [×4], C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C2×D4⋊S3 [×2], C322D8 [×2], C2×D6⋊S3, C2×C322D8

Smallest permutation representation of C2×C322D8
On 96 points
Generators in S96
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 87)(10 88)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 72)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)
(1 13 94)(2 95 14)(3 15 96)(4 89 16)(5 9 90)(6 91 10)(7 11 92)(8 93 12)(17 79 32)(18 25 80)(19 73 26)(20 27 74)(21 75 28)(22 29 76)(23 77 30)(24 31 78)(33 58 66)(34 67 59)(35 60 68)(36 69 61)(37 62 70)(38 71 63)(39 64 72)(40 65 57)(41 51 88)(42 81 52)(43 53 82)(44 83 54)(45 55 84)(46 85 56)(47 49 86)(48 87 50)
(1 13 94)(2 95 14)(3 15 96)(4 89 16)(5 9 90)(6 91 10)(7 11 92)(8 93 12)(17 32 79)(18 80 25)(19 26 73)(20 74 27)(21 28 75)(22 76 29)(23 30 77)(24 78 31)(33 66 58)(34 59 67)(35 68 60)(36 61 69)(37 70 62)(38 63 71)(39 72 64)(40 57 65)(41 51 88)(42 81 52)(43 53 82)(44 83 54)(45 55 84)(46 85 56)(47 49 86)(48 87 50)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(33 50)(34 49)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 57)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 91)(74 90)(75 89)(76 96)(77 95)(78 94)(79 93)(80 92)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 S3 D4 D4 D6 D6 D8 C3⋊D4 C3⋊D4 S32 D4⋊S3 D6⋊S3 C2×S32 D6⋊S3 C32⋊2D8 kernel C2×C32⋊2D8 C32⋊2D8 C2×C32⋊4C8 C6×D12 C2×D12 C3×C12 C62 D12 C2×C12 C3×C6 C12 C2×C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 2 2 1 1 4 2 4 4 4 1 4 1 1 1 4

Matrix representation of C2×C322D8 in GL6(𝔽73)

 72 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 0 0 0 0 0 0 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 0 1 0 0 0 0 0 0 1
,
 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 72 1 0 0 0 0 72 0
,
 51 48 0 0 0 0 0 63 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 36 48 0 0 0 0 11 37
,
 6 1 0 0 0 0 38 67 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 50 0 0 0 0 68 55

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[51,0,0,0,0,0,48,63,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,11,0,0,0,0,48,37],[6,38,0,0,0,0,1,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,68,0,0,0,0,50,55] >;

C2×C322D8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2D_8
% in TeX

G:=Group("C2xC3^2:2D8");
// GroupNames label

G:=SmallGroup(288,469);
// by ID

G=gap.SmallGroup(288,469);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,675,346,80,1356,9414]);
// Polycyclic

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

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