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## G = C3×C8⋊7D4order 192 = 26·3

### Direct product of C3 and C8⋊7D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8⋊7D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×D4 — C6×D8 — C3×C8⋊7D4
 Lower central C1 — C2 — C2×C4 — C3×C8⋊7D4
 Upper central C1 — C2×C6 — C22×C12 — C3×C8⋊7D4

Generators and relations for C3×C87D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 282 in 134 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×D8, C22×C12, C6×D4, C6×D4, C87D4, C3×D4⋊C4, C3×C2.D8, C3×C4⋊D4, C22×C24, C6×D8, C3×C87D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C2×D8, C4○D8, C3×D8, C6×D4, C3×C4○D4, C87D4, C3×C4⋊D4, C6×D8, C3×C4○D8, C3×C87D4

Smallest permutation representation of C3×C87D4
On 96 points
Generators in S96
(1 83 33)(2 84 34)(3 85 35)(4 86 36)(5 87 37)(6 88 38)(7 81 39)(8 82 40)(9 62 25)(10 63 26)(11 64 27)(12 57 28)(13 58 29)(14 59 30)(15 60 31)(16 61 32)(17 78 67)(18 79 68)(19 80 69)(20 73 70)(21 74 71)(22 75 72)(23 76 65)(24 77 66)(41 50 91)(42 51 92)(43 52 93)(44 53 94)(45 54 95)(46 55 96)(47 56 89)(48 49 90)
(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 23 13 48)(2 22 14 47)(3 21 15 46)(4 20 16 45)(5 19 9 44)(6 18 10 43)(7 17 11 42)(8 24 12 41)(25 94 37 69)(26 93 38 68)(27 92 39 67)(28 91 40 66)(29 90 33 65)(30 89 34 72)(31 96 35 71)(32 95 36 70)(49 83 76 58)(50 82 77 57)(51 81 78 64)(52 88 79 63)(53 87 80 62)(54 86 73 61)(55 85 74 60)(56 84 75 59)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 22)(18 21)(19 20)(23 24)(25 36)(26 35)(27 34)(28 33)(29 40)(30 39)(31 38)(32 37)(41 48)(42 47)(43 46)(44 45)(49 50)(51 56)(52 55)(53 54)(57 83)(58 82)(59 81)(60 88)(61 87)(62 86)(63 85)(64 84)(65 66)(67 72)(68 71)(69 70)(73 80)(74 79)(75 78)(76 77)(89 92)(90 91)(93 96)(94 95)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 8A ··· 8H 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 8 8 1 1 2 2 2 2 8 8 1 ··· 1 2 2 2 2 8 8 8 8 2 ··· 2 2 ··· 2 8 8 8 8 2 ··· 2

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D4 C4○D4 D8 C3×D4 C3×D4 C3×D4 C4○D8 C3×C4○D4 C3×D8 C3×C4○D8 kernel C3×C8⋊7D4 C3×D4⋊C4 C3×C2.D8 C3×C4⋊D4 C22×C24 C6×D8 C8⋊7D4 D4⋊C4 C2.D8 C4⋊D4 C22×C8 C2×D8 C24 C2×C12 C22×C6 C12 C2×C6 C8 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 2 1 1 2 4 4 2 2 4 4 8 8

Matrix representation of C3×C87D4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 0 41 0 0 16 41 0 0 0 0 57 16 0 0 57 57
,
 0 32 0 0 16 0 0 0 0 0 6 6 0 0 6 67
,
 0 41 0 0 57 0 0 0 0 0 57 16 0 0 16 16
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,41,41,0,0,0,0,57,57,0,0,16,57],[0,16,0,0,32,0,0,0,0,0,6,6,0,0,6,67],[0,57,0,0,41,0,0,0,0,0,57,16,0,0,16,16] >;

C3×C87D4 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_7D_4
% in TeX

G:=Group("C3xC8:7D4");
// GroupNames label

G:=SmallGroup(192,899);
// by ID

G=gap.SmallGroup(192,899);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,4204,172]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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