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## G = C4×C32⋊2Q8order 288 = 25·32

### Direct product of C4 and C32⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C4×C32⋊2Q8
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C32⋊2Q8 — C4×C32⋊2Q8
 Lower central C32 — C3×C6 — C4×C32⋊2Q8
 Upper central C1 — C2×C4

Generators and relations for C4×C322Q8
G = < a,b,c,d,e | a4=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 458 in 155 conjugacy classes, 66 normal (18 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×9], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×4], Dic3 [×12], C12 [×4], C12 [×8], C2×C6 [×2], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], Dic6 [×8], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C4×Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C62, C4×Dic3 [×2], C4×Dic3 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4×C12 [×2], C2×Dic6 [×2], C322Q8 [×4], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C4×Dic6 [×2], Dic3⋊Dic3 [×2], C62.C22, Dic3×C12 [×2], C4×C3⋊Dic3, C2×C322Q8, C4×C322Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], Q8 [×2], C23, D6 [×6], C22×C4, C2×Q8, C4○D4, Dic6 [×4], C4×S3 [×4], C22×S3 [×2], C4×Q8, S32, C2×Dic6 [×2], S3×C2×C4 [×2], C4○D12 [×2], C322Q8 [×2], C2×S32, C4×Dic6 [×2], D6.D6, C4×S32, C2×C322Q8, C4×C322Q8

Smallest permutation representation of C4×C322Q8
On 96 points
Generators in S96
(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 37 23)(2 38 24)(3 39 21)(4 40 22)(5 94 84)(6 95 81)(7 96 82)(8 93 83)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 17 91)(14 18 92)(15 19 89)(16 20 90)(33 61 47)(34 62 48)(35 63 45)(36 64 46)(41 52 54)(42 49 55)(43 50 56)(44 51 53)(57 71 85)(58 72 86)(59 69 87)(60 70 88)(65 78 76)(66 79 73)(67 80 74)(68 77 75)
(1 37 23)(2 38 24)(3 39 21)(4 40 22)(5 94 84)(6 95 81)(7 96 82)(8 93 83)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 17 91)(14 18 92)(15 19 89)(16 20 90)(33 47 61)(34 48 62)(35 45 63)(36 46 64)(41 54 52)(42 55 49)(43 56 50)(44 53 51)(57 85 71)(58 86 72)(59 87 69)(60 88 70)(65 76 78)(66 73 79)(67 74 80)(68 75 77)
(1 51 27 47)(2 52 28 48)(3 49 25 45)(4 50 26 46)(5 67 91 88)(6 68 92 85)(7 65 89 86)(8 66 90 87)(9 61 37 44)(10 62 38 41)(11 63 39 42)(12 64 40 43)(13 70 94 74)(14 71 95 75)(15 72 96 76)(16 69 93 73)(17 60 84 80)(18 57 81 77)(19 58 82 78)(20 59 83 79)(21 55 31 35)(22 56 32 36)(23 53 29 33)(24 54 30 34)
(1 73 27 69)(2 74 28 70)(3 75 25 71)(4 76 26 72)(5 62 91 41)(6 63 92 42)(7 64 89 43)(8 61 90 44)(9 87 37 66)(10 88 38 67)(11 85 39 68)(12 86 40 65)(13 52 94 48)(14 49 95 45)(15 50 96 46)(16 51 93 47)(17 54 84 34)(18 55 81 35)(19 56 82 36)(20 53 83 33)(21 77 31 57)(22 78 32 58)(23 79 29 59)(24 80 30 60)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I 12J 12K 12L 12M ··· 12AB order 1 2 2 2 3 3 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 4 1 1 1 1 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + - + - + image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D6 D6 C4○D4 C4×S3 Dic6 C4○D12 S32 C32⋊2Q8 C2×S32 D6.D6 C4×S32 kernel C4×C32⋊2Q8 Dic3⋊Dic3 C62.C22 Dic3×C12 C4×C3⋊Dic3 C2×C32⋊2Q8 C32⋊2Q8 C4×Dic3 C3×C12 C2×Dic3 C2×C12 C3×C6 Dic3 C12 C6 C2×C4 C4 C22 C2 C2 # reps 1 2 1 2 1 1 8 2 2 4 2 2 8 8 8 1 2 1 2 2

Matrix representation of C4×C322Q8 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 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 0 12 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C4×C322Q8 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C4xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(288,565);
// by ID

G=gap.SmallGroup(288,565);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,58,1356,9414]);
// Polycyclic

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

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