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## G = C4×C7⋊C12order 336 = 24·3·7

### Direct product of C4 and C7⋊C12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C4×C7⋊C12
 Chief series C1 — C7 — C14 — C2×C14 — C22×C7⋊C3 — C2×C7⋊C12 — C4×C7⋊C12
 Lower central C7 — C4×C7⋊C12
 Upper central C1 — C2×C4

Generators and relations for C4×C7⋊C12
G = < a,b,c | a4=b7=c12=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 192 in 60 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C7, C2×C4, C2×C4, C12, C2×C6, C14, C14, C42, C7⋊C3, C2×C12, Dic7, C28, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C4×C12, C2×Dic7, C2×C28, C7⋊C12, C4×C7⋊C3, C22×C7⋊C3, C4×Dic7, C2×C7⋊C12, C2×C4×C7⋊C3, C4×C7⋊C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, C2×C12, F7, C4×C12, C7⋊C12, C2×F7, C4×F7, C2×C7⋊C12, C4×C7⋊C12

Smallest permutation representation of C4×C7⋊C12
On 112 points
Generators in S112
(1 12 6 14)(2 9 7 15)(3 10 8 16)(4 11 5 13)(17 49 85 105)(18 50 86 106)(19 51 87 107)(20 52 88 108)(21 41 77 109)(22 42 78 110)(23 43 79 111)(24 44 80 112)(25 45 81 101)(26 46 82 102)(27 47 83 103)(28 48 84 104)(29 75 57 91)(30 76 58 92)(31 65 59 93)(32 66 60 94)(33 67 61 95)(34 68 62 96)(35 69 63 97)(36 70 64 98)(37 71 53 99)(38 72 54 100)(39 73 55 89)(40 74 56 90)
(1 98 94 26 90 18 22)(2 19 27 99 23 91 95)(3 92 100 20 96 24 28)(4 25 21 93 17 97 89)(5 81 77 65 85 69 73)(6 70 66 82 74 86 78)(7 87 83 71 79 75 67)(8 76 72 88 68 80 84)(9 51 47 37 43 29 33)(10 30 38 52 34 44 48)(11 45 41 31 49 35 39)(12 36 32 46 40 50 42)(13 101 109 59 105 63 55)(14 64 60 102 56 106 110)(15 107 103 53 111 57 61)(16 58 54 108 62 112 104)
(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 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112)

G:=sub<Sym(112)| (1,12,6,14)(2,9,7,15)(3,10,8,16)(4,11,5,13)(17,49,85,105)(18,50,86,106)(19,51,87,107)(20,52,88,108)(21,41,77,109)(22,42,78,110)(23,43,79,111)(24,44,80,112)(25,45,81,101)(26,46,82,102)(27,47,83,103)(28,48,84,104)(29,75,57,91)(30,76,58,92)(31,65,59,93)(32,66,60,94)(33,67,61,95)(34,68,62,96)(35,69,63,97)(36,70,64,98)(37,71,53,99)(38,72,54,100)(39,73,55,89)(40,74,56,90), (1,98,94,26,90,18,22)(2,19,27,99,23,91,95)(3,92,100,20,96,24,28)(4,25,21,93,17,97,89)(5,81,77,65,85,69,73)(6,70,66,82,74,86,78)(7,87,83,71,79,75,67)(8,76,72,88,68,80,84)(9,51,47,37,43,29,33)(10,30,38,52,34,44,48)(11,45,41,31,49,35,39)(12,36,32,46,40,50,42)(13,101,109,59,105,63,55)(14,64,60,102,56,106,110)(15,107,103,53,111,57,61)(16,58,54,108,62,112,104), (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)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112)>;

G:=Group( (1,12,6,14)(2,9,7,15)(3,10,8,16)(4,11,5,13)(17,49,85,105)(18,50,86,106)(19,51,87,107)(20,52,88,108)(21,41,77,109)(22,42,78,110)(23,43,79,111)(24,44,80,112)(25,45,81,101)(26,46,82,102)(27,47,83,103)(28,48,84,104)(29,75,57,91)(30,76,58,92)(31,65,59,93)(32,66,60,94)(33,67,61,95)(34,68,62,96)(35,69,63,97)(36,70,64,98)(37,71,53,99)(38,72,54,100)(39,73,55,89)(40,74,56,90), (1,98,94,26,90,18,22)(2,19,27,99,23,91,95)(3,92,100,20,96,24,28)(4,25,21,93,17,97,89)(5,81,77,65,85,69,73)(6,70,66,82,74,86,78)(7,87,83,71,79,75,67)(8,76,72,88,68,80,84)(9,51,47,37,43,29,33)(10,30,38,52,34,44,48)(11,45,41,31,49,35,39)(12,36,32,46,40,50,42)(13,101,109,59,105,63,55)(14,64,60,102,56,106,110)(15,107,103,53,111,57,61)(16,58,54,108,62,112,104), (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)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112) );

G=PermutationGroup([[(1,12,6,14),(2,9,7,15),(3,10,8,16),(4,11,5,13),(17,49,85,105),(18,50,86,106),(19,51,87,107),(20,52,88,108),(21,41,77,109),(22,42,78,110),(23,43,79,111),(24,44,80,112),(25,45,81,101),(26,46,82,102),(27,47,83,103),(28,48,84,104),(29,75,57,91),(30,76,58,92),(31,65,59,93),(32,66,60,94),(33,67,61,95),(34,68,62,96),(35,69,63,97),(36,70,64,98),(37,71,53,99),(38,72,54,100),(39,73,55,89),(40,74,56,90)], [(1,98,94,26,90,18,22),(2,19,27,99,23,91,95),(3,92,100,20,96,24,28),(4,25,21,93,17,97,89),(5,81,77,65,85,69,73),(6,70,66,82,74,86,78),(7,87,83,71,79,75,67),(8,76,72,88,68,80,84),(9,51,47,37,43,29,33),(10,30,38,52,34,44,48),(11,45,41,31,49,35,39),(12,36,32,46,40,50,42),(13,101,109,59,105,63,55),(14,64,60,102,56,106,110),(15,107,103,53,111,57,61),(16,58,54,108,62,112,104)], [(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),(89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112)]])

56 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 7 12A ··· 12X 14A 14B 14C 28A 28B 28C 28D order 1 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 7 12 ··· 12 14 14 14 28 28 28 28 size 1 1 1 1 7 7 1 1 1 1 7 ··· 7 7 ··· 7 6 7 ··· 7 6 6 6 6 6 6 6

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 6 6 6 6 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 F7 C7⋊C12 C2×F7 C4×F7 kernel C4×C7⋊C12 C2×C7⋊C12 C2×C4×C7⋊C3 C4×Dic7 C7⋊C12 C4×C7⋊C3 C2×Dic7 C2×C28 Dic7 C28 C2×C4 C4 C22 C2 # reps 1 2 1 2 8 4 4 2 16 8 1 2 1 4

Matrix representation of C4×C7⋊C12 in GL7(𝔽337)

 1 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 336 0 1 0 0 0 0 336 0 0 1 0 0 0 336 0 0 0 1 0 0 336 0 0 0 0 1 0 336 0 0 0 0 0 1 336
,
 265 0 0 0 0 0 0 0 8 0 0 8 240 329 0 8 8 240 0 329 0 0 248 0 329 8 329 0 0 0 8 329 8 0 240 0 0 8 0 248 329 329 0 8 248 329 0 0 329

G:=sub<GL(7,GF(337))| [1,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,336,336,336,336,336,336],[265,0,0,0,0,0,0,0,8,8,248,0,0,8,0,0,8,0,8,8,248,0,0,240,329,329,0,329,0,8,0,8,8,248,0,0,240,329,329,0,329,0,0,329,0,0,240,329,329] >;

C4×C7⋊C12 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes C_{12}
% in TeX

G:=Group("C4xC7:C12");
// GroupNames label

G:=SmallGroup(336,14);
// by ID

G=gap.SmallGroup(336,14);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,151,10373,1745]);
// Polycyclic

G:=Group<a,b,c|a^4=b^7=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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