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## G = C2×C62.C22order 288 = 25·32

### Direct product of C2 and C62.C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C62.C22
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C62.C22 — C2×C62.C22
 Lower central C32 — C3×C6 — C2×C62.C22
 Upper central C1 — C23

Generators and relations for C2×C62.C22
G = < a,b,c,d,e | a2=b6=c6=1, d2=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=c3d >

Subgroups: 594 in 211 conjugacy classes, 92 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×8], C22, C22 [×6], C6 [×14], C6 [×7], C2×C4 [×14], C23, C32, Dic3 [×16], C12 [×4], C2×C6 [×14], C2×C6 [×7], C4⋊C4 [×4], C22×C4 [×3], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×4], C2×Dic3 [×22], C2×C12 [×8], C22×C6 [×2], C22×C6, C2×C4⋊C4, C3×Dic3 [×4], C3⋊Dic3 [×4], C62, C62 [×6], Dic3⋊C4 [×8], C22×Dic3 [×2], C22×Dic3 [×3], C22×C12 [×2], C6×Dic3 [×4], C6×Dic3 [×4], C2×C3⋊Dic3 [×6], C2×C62, C2×Dic3⋊C4 [×2], C62.C22 [×4], Dic3×C2×C6 [×2], C22×C3⋊Dic3, C2×C62.C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×4], C4×S3 [×4], C3⋊D4 [×4], C22×S3 [×2], C2×C4⋊C4, S32, Dic3⋊C4 [×8], C2×Dic6 [×2], S3×C2×C4 [×2], C2×C3⋊D4 [×2], C6.D6 [×2], D6⋊S3 [×2], C322Q8 [×2], C2×S32, C2×Dic3⋊C4 [×2], C62.C22 [×4], C2×C6.D6, C2×D6⋊S3, C2×C322Q8, C2×C62.C22

Smallest permutation representation of C2×C62.C22
On 96 points
Generators in S96
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 89)(8 90)(9 85)(10 86)(11 87)(12 88)(13 27)(14 28)(15 29)(16 30)(17 25)(18 26)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 53)(38 54)(39 49)(40 50)(41 51)(42 52)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(61 75)(62 76)(63 77)(64 78)(65 73)(66 74)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(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 18 5 16 3 14)(2 13 6 17 4 15)(7 95 11 93 9 91)(8 96 12 94 10 92)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)(43 53 45 49 47 51)(44 54 46 50 48 52)(55 63 57 65 59 61)(56 64 58 66 60 62)(67 74 69 76 71 78)(68 75 70 77 72 73)(79 86 83 90 81 88)(80 87 84 85 82 89)
(1 69 4 72)(2 70 5 67)(3 71 6 68)(7 38 10 41)(8 39 11 42)(9 40 12 37)(13 75 16 78)(14 76 17 73)(15 77 18 74)(19 58 22 55)(20 59 23 56)(21 60 24 57)(25 65 28 62)(26 66 29 63)(27 61 30 64)(31 94 34 91)(32 95 35 92)(33 96 36 93)(43 82 46 79)(44 83 47 80)(45 84 48 81)(49 87 52 90)(50 88 53 85)(51 89 54 86)
(1 45 4 48)(2 44 5 47)(3 43 6 46)(7 57 10 60)(8 56 11 59)(9 55 12 58)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 32 22 35)(20 31 23 34)(21 36 24 33)(25 40 28 37)(26 39 29 42)(27 38 30 41)(61 96 64 93)(62 95 65 92)(63 94 66 91)(67 85 70 88)(68 90 71 87)(69 89 72 86)(73 83 76 80)(74 82 77 79)(75 81 78 84)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 4A ··· 4H 4I 4J 4K 4L 6A ··· 6N 6O ··· 6U 12A ··· 12P order 1 2 ··· 2 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 4 6 ··· 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + - + + - + + - - + image C1 C2 C2 C2 C4 S3 D4 Q8 D6 D6 Dic6 C4×S3 C3⋊D4 S32 C6.D6 D6⋊S3 C32⋊2Q8 C2×S32 kernel C2×C62.C22 C62.C22 Dic3×C2×C6 C22×C3⋊Dic3 C2×C3⋊Dic3 C22×Dic3 C62 C62 C2×Dic3 C22×C6 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C22 # reps 1 4 2 1 8 2 2 2 4 2 8 8 8 1 2 2 2 1

Matrix representation of C2×C62.C22 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 6 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 11 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,4,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C2×C62.C22 in GAP, Magma, Sage, TeX

C_2\times C_6^2.C_2^2
% in TeX

G:=Group("C2xC6^2.C2^2");
// GroupNames label

G:=SmallGroup(288,615);
// by ID

G=gap.SmallGroup(288,615);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,64,1356,9414]);
// Polycyclic

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

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