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## G = C22×C32⋊2Q8order 288 = 25·32

### Direct product of C22 and C32⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C22×C32⋊2Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3×Dic3 — C32⋊2Q8 — C2×C32⋊2Q8 — C22×C32⋊2Q8
 Lower central C32 — C3×C6 — C22×C32⋊2Q8
 Upper central C1 — C23

Generators and relations for C22×C322Q8
G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=c-1, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 978 in 339 conjugacy classes, 148 normal (8 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×12], C22 [×7], C6 [×14], C6 [×7], C2×C4 [×18], Q8 [×16], C23, C32, Dic3 [×8], Dic3 [×12], C12 [×8], C2×C6 [×14], C2×C6 [×7], C22×C4 [×3], C2×Q8 [×12], C3×C6, C3×C6 [×6], Dic6 [×32], C2×Dic3 [×12], C2×Dic3 [×18], C2×C12 [×12], C22×C6 [×2], C22×C6, C22×Q8, C3×Dic3 [×8], C3⋊Dic3 [×4], C62 [×7], C2×Dic6 [×24], C22×Dic3 [×2], C22×Dic3 [×3], C22×C12 [×2], C322Q8 [×16], C6×Dic3 [×12], C2×C3⋊Dic3 [×6], C2×C62, C22×Dic6 [×2], C2×C322Q8 [×12], Dic3×C2×C6 [×2], C22×C3⋊Dic3, C22×C322Q8
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], Q8 [×4], C23 [×15], D6 [×14], C2×Q8 [×6], C24, Dic6 [×8], C22×S3 [×14], C22×Q8, S32, C2×Dic6 [×12], S3×C23 [×2], C322Q8 [×4], C2×S32 [×3], C22×Dic6 [×2], C2×C322Q8 [×6], C22×S32, C22×C322Q8

Smallest permutation representation of C22×C322Q8
On 96 points
Generators in S96
(1 33)(2 34)(3 35)(4 36)(5 47)(6 48)(7 45)(8 46)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)(25 29)(26 30)(27 31)(28 32)(37 91)(38 92)(39 89)(40 90)(41 64)(42 61)(43 62)(44 63)(49 78)(50 79)(51 80)(52 77)(53 95)(54 96)(55 93)(56 94)(57 86)(58 87)(59 88)(60 85)(65 83)(66 84)(67 81)(68 82)(69 76)(70 73)(71 74)(72 75)
(1 11)(2 12)(3 9)(4 10)(5 17)(6 18)(7 19)(8 20)(13 48)(14 45)(15 46)(16 47)(21 35)(22 36)(23 33)(24 34)(25 62)(26 63)(27 64)(28 61)(29 43)(30 44)(31 41)(32 42)(37 77)(38 78)(39 79)(40 80)(49 92)(50 89)(51 90)(52 91)(53 81)(54 82)(55 83)(56 84)(57 75)(58 76)(59 73)(60 74)(65 93)(66 94)(67 95)(68 96)(69 87)(70 88)(71 85)(72 86)
(1 27 14)(2 15 28)(3 25 16)(4 13 26)(5 21 43)(6 44 22)(7 23 41)(8 42 24)(9 62 47)(10 48 63)(11 64 45)(12 46 61)(17 35 29)(18 30 36)(19 33 31)(20 32 34)(37 95 76)(38 73 96)(39 93 74)(40 75 94)(49 88 82)(50 83 85)(51 86 84)(52 81 87)(53 69 91)(54 92 70)(55 71 89)(56 90 72)(57 66 80)(58 77 67)(59 68 78)(60 79 65)
(1 14 27)(2 15 28)(3 16 25)(4 13 26)(5 43 21)(6 44 22)(7 41 23)(8 42 24)(9 47 62)(10 48 63)(11 45 64)(12 46 61)(17 29 35)(18 30 36)(19 31 33)(20 32 34)(37 95 76)(38 96 73)(39 93 74)(40 94 75)(49 82 88)(50 83 85)(51 84 86)(52 81 87)(53 69 91)(54 70 92)(55 71 89)(56 72 90)(57 80 66)(58 77 67)(59 78 68)(60 79 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 69 3 71)(2 72 4 70)(5 65 7 67)(6 68 8 66)(9 85 11 87)(10 88 12 86)(13 54 15 56)(14 53 16 55)(17 93 19 95)(18 96 20 94)(21 60 23 58)(22 59 24 57)(25 89 27 91)(26 92 28 90)(29 39 31 37)(30 38 32 40)(33 76 35 74)(34 75 36 73)(41 77 43 79)(42 80 44 78)(45 81 47 83)(46 84 48 82)(49 61 51 63)(50 64 52 62)

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

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

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

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 2 2 2 2 2 4 4 4 type + + + + + - + + - + - + image C1 C2 C2 C2 S3 Q8 D6 D6 Dic6 S32 C32⋊2Q8 C2×S32 kernel C22×C32⋊2Q8 C2×C32⋊2Q8 Dic3×C2×C6 C22×C3⋊Dic3 C22×Dic3 C62 C2×Dic3 C22×C6 C2×C6 C23 C22 C22 # reps 1 12 2 1 2 4 12 2 16 1 4 3

Matrix representation of C22×C322Q8 in GL8(𝔽13)

 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 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 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
,
 1 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 1 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
,
 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 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
,
 8 2 0 0 0 0 0 0 0 5 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 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 4 5 0 0 0 0 0 0 7 9 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 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

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[4,7,0,0,0,0,0,0,5,9,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C22×C322Q8 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C2^2xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(288,975);
// by ID

G=gap.SmallGroup(288,975);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,1356,9414]);
// Polycyclic

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

׿
×
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