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## G = C42×C7⋊C3order 336 = 24·3·7

### Direct product of C42 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C42×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C14 — C22×C7⋊C3 — C2×C4×C7⋊C3 — C42×C7⋊C3
 Lower central C7 — C42×C7⋊C3
 Upper central C1 — C42

Generators and relations for C42×C7⋊C3
G = < a,b,c,d | a4=b4=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 150 in 60 conjugacy classes, 45 normal (9 characteristic)
C1, C2, C3, C4, C22, C6, C7, C2×C4, C12, C2×C6, C14, C42, C7⋊C3, C2×C12, C28, C2×C14, C2×C7⋊C3, C4×C12, C2×C28, C4×C7⋊C3, C22×C7⋊C3, C4×C28, C2×C4×C7⋊C3, C42×C7⋊C3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, C7⋊C3, C2×C12, C2×C7⋊C3, C4×C12, C4×C7⋊C3, C22×C7⋊C3, C2×C4×C7⋊C3, C42×C7⋊C3

Smallest permutation representation of C42×C7⋊C3
On 112 points
Generators in S112
(1 99 22 92)(2 100 23 93)(3 101 24 94)(4 102 25 95)(5 103 26 96)(6 104 27 97)(7 105 28 98)(8 106 15 85)(9 107 16 86)(10 108 17 87)(11 109 18 88)(12 110 19 89)(13 111 20 90)(14 112 21 91)(29 57 50 78)(30 58 51 79)(31 59 52 80)(32 60 53 81)(33 61 54 82)(34 62 55 83)(35 63 56 84)(36 64 43 71)(37 65 44 72)(38 66 45 73)(39 67 46 74)(40 68 47 75)(41 69 48 76)(42 70 49 77)
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)(58 59 61)(60 63 62)(65 66 68)(67 70 69)(72 73 75)(74 77 76)(79 80 82)(81 84 83)(86 87 89)(88 91 90)(93 94 96)(95 98 97)(100 101 103)(102 105 104)(107 108 110)(109 112 111)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C4 C6 C12 C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 kernel C42×C7⋊C3 C2×C4×C7⋊C3 C4×C28 C4×C7⋊C3 C2×C28 C28 C42 C2×C4 C4 # reps 1 3 2 12 6 24 2 6 24

Matrix representation of C42×C7⋊C3 in GL5(𝔽337)

 148 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 336 0 0 0 0 0 148 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 124 125 1 0 0 1 0 0 0 0 0 1 0
,
 208 0 0 0 0 0 208 0 0 0 0 0 1 0 0 0 0 212 336 336 0 0 0 1 0

G:=sub<GL(5,GF(337))| [148,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[336,0,0,0,0,0,148,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,124,1,0,0,0,125,0,1,0,0,1,0,0],[208,0,0,0,0,0,208,0,0,0,0,0,1,212,0,0,0,0,336,1,0,0,0,336,0] >;

C42×C7⋊C3 in GAP, Magma, Sage, TeX

C_4^2\times C_7\rtimes C_3
% in TeX

G:=Group("C4^2xC7:C3");
// GroupNames label

G:=SmallGroup(336,48);
// by ID

G=gap.SmallGroup(336,48);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,79,881]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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