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## G = C22×C4×C7⋊C3order 336 = 24·3·7

### Direct product of C22×C4 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C22×C4×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C22×C7⋊C3 — C23×C7⋊C3 — C22×C4×C7⋊C3
 Lower central C7 — C22×C4×C7⋊C3
 Upper central C1 — C22×C4

Generators and relations for C22×C4×C7⋊C3
G = < a,b,c,d,e | a2=b2=c4=d7=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 270 in 108 conjugacy classes, 81 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C7, C2×C4, C23, C12, C2×C6, C14, C14, C22×C4, C7⋊C3, C2×C12, C22×C6, C28, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C22×C12, C2×C28, C22×C14, C4×C7⋊C3, C22×C7⋊C3, C22×C28, C2×C4×C7⋊C3, C23×C7⋊C3, C22×C4×C7⋊C3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C7⋊C3, C2×C12, C22×C6, C2×C7⋊C3, C22×C12, C4×C7⋊C3, C22×C7⋊C3, C2×C4×C7⋊C3, C23×C7⋊C3, C22×C4×C7⋊C3

Smallest permutation representation of C22×C4×C7⋊C3
On 112 points
Generators in S112
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 15 8 22)(2 16 9 23)(3 17 10 24)(4 18 11 25)(5 19 12 26)(6 20 13 27)(7 21 14 28)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)
(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)| (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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( (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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([[(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(1,15,8,22),(2,16,9,23),(3,17,10,24),(4,18,11,25),(5,19,12,26),(6,20,13,27),(7,21,14,28),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56),(57,71,64,78),(58,72,65,79),(59,73,66,80),(60,74,67,81),(61,75,68,82),(62,76,69,83),(63,77,70,84),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112)], [(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 ··· 2G 3A 3B 4A ··· 4H 6A ··· 6N 7A 7B 12A ··· 12P 14A ··· 14N 28A ··· 28P order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 7 7 12 ··· 12 14 ··· 14 28 ··· 28 size 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 1 1 3 3 3 3 type + + + image C1 C2 C2 C3 C4 C6 C6 C12 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 kernel C22×C4×C7⋊C3 C2×C4×C7⋊C3 C23×C7⋊C3 C22×C28 C22×C7⋊C3 C2×C28 C22×C14 C2×C14 C22×C4 C2×C4 C23 C22 # reps 1 6 1 2 8 12 2 16 2 12 2 16

Matrix representation of C22×C4×C7⋊C3 in GL5(𝔽337)

 336 0 0 0 0 0 1 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 336 0 0 0 0 0 336 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 148 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 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
,
 128 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))| [336,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[336,0,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[148,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[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],[128,0,0,0,0,0,208,0,0,0,0,0,1,212,0,0,0,0,336,1,0,0,0,336,0] >;

C22×C4×C7⋊C3 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times C_7\rtimes C_3
% in TeX

G:=Group("C2^2xC4xC7:C3");
// GroupNames label

G:=SmallGroup(336,164);
// by ID

G=gap.SmallGroup(336,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,122,455]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^7=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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