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## G = C4⋊C4×C7⋊C3order 336 = 24·3·7

### Direct product of C4⋊C4 and C7⋊C3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊C4×C7⋊C3
G = < a,b,c,d | a4=b4=c7=d3=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of C4⋊C4×C7⋊C3
On 112 points
Generators in S112
(1 78 8 71)(2 79 9 72)(3 80 10 73)(4 81 11 74)(5 82 12 75)(6 83 13 76)(7 84 14 77)(15 64 22 57)(16 65 23 58)(17 66 24 59)(18 67 25 60)(19 68 26 61)(20 69 27 62)(21 70 28 63)(29 106 36 99)(30 107 37 100)(31 108 38 101)(32 109 39 102)(33 110 40 103)(34 111 41 104)(35 112 42 105)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)
(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)(57 106 71 92)(58 107 72 93)(59 108 73 94)(60 109 74 95)(61 110 75 96)(62 111 76 97)(63 112 77 98)(64 99 78 85)(65 100 79 86)(66 101 80 87)(67 102 81 88)(68 103 82 89)(69 104 83 90)(70 105 84 91)
(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)| (1,78,8,71)(2,79,9,72)(3,80,10,73)(4,81,11,74)(5,82,12,75)(6,83,13,76)(7,84,14,77)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (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)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91), (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( (1,78,8,71)(2,79,9,72)(3,80,10,73)(4,81,11,74)(5,82,12,75)(6,83,13,76)(7,84,14,77)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (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)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91), (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([[(1,78,8,71),(2,79,9,72),(3,80,10,73),(4,81,11,74),(5,82,12,75),(6,83,13,76),(7,84,14,77),(15,64,22,57),(16,65,23,58),(17,66,24,59),(18,67,25,60),(19,68,26,61),(20,69,27,62),(21,70,28,63),(29,106,36,99),(30,107,37,100),(31,108,38,101),(32,109,39,102),(33,110,40,103),(34,111,41,104),(35,112,42,105),(43,92,50,85),(44,93,51,86),(45,94,52,87),(46,95,53,88),(47,96,54,89),(48,97,55,90),(49,98,56,91)], [(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),(57,106,71,92),(58,107,72,93),(59,108,73,94),(60,109,74,95),(61,110,75,96),(62,111,76,97),(63,112,77,98),(64,99,78,85),(65,100,79,86),(66,101,80,87),(67,102,81,88),(68,103,82,89),(69,104,83,90),(70,105,84,91)], [(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)]])

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + - image C1 C2 C3 C4 C6 C12 D4 Q8 C3×D4 C3×Q8 C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 D4×C7⋊C3 Q8×C7⋊C3 kernel C4⋊C4×C7⋊C3 C2×C4×C7⋊C3 C7×C4⋊C4 C4×C7⋊C3 C2×C28 C28 C2×C7⋊C3 C2×C7⋊C3 C14 C14 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 3 2 4 6 8 1 1 2 2 2 6 8 2 2

Matrix representation of C4⋊C4×C7⋊C3 in GL7(𝔽337)

 12 314 0 0 0 0 0 314 325 0 0 0 0 0 0 0 325 23 0 0 0 0 0 23 12 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 1 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 148 0 0 0 0 0 189 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336
,
 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 0 0 1 0 0 0 0 1 0 213 0 0 0 0 0 1 212
,
 208 0 0 0 0 0 0 0 208 0 0 0 0 0 0 0 128 0 0 0 0 0 0 0 128 0 0 0 0 0 0 0 1 0 212 0 0 0 0 0 0 336 0 0 0 0 0 1 336

G:=sub<GL(7,GF(337))| [12,314,0,0,0,0,0,314,325,0,0,0,0,0,0,0,325,23,0,0,0,0,0,23,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,336,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,189,0,0,0,0,0,148,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,213,212],[208,0,0,0,0,0,0,0,208,0,0,0,0,0,0,0,128,0,0,0,0,0,0,0,128,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,212,336,336] >;

C4⋊C4×C7⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("C4:C4xC7:C3");
// GroupNames label

G:=SmallGroup(336,50);
// by ID

G=gap.SmallGroup(336,50);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,313,79,881]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^7=d^3=1,b*a*b^-1=a^-1,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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