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## G = C22×C7⋊C12order 336 = 24·3·7

### Direct product of C22 and C7⋊C12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C22×C7⋊C12
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C7⋊C12 — C2×C7⋊C12 — C22×C7⋊C12
 Lower central C7 — C22×C7⋊C12
 Upper central C1 — C23

Generators and relations for C22×C7⋊C12
G = < a,b,c,d | a2=b2=c7=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 336 in 108 conjugacy classes, 70 normal (11 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, Dic7, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C22×C12, C2×Dic7, C22×C14, C7⋊C12, C22×C7⋊C3, C22×Dic7, C2×C7⋊C12, C23×C7⋊C3, C22×C7⋊C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, F7, C22×C12, C7⋊C12, C2×F7, C2×C7⋊C12, C22×F7, C22×C7⋊C12

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 6 6 6 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 F7 C7⋊C12 C2×F7 kernel C22×C7⋊C12 C2×C7⋊C12 C23×C7⋊C3 C22×Dic7 C22×C7⋊C3 C2×Dic7 C22×C14 C2×C14 C23 C22 C22 # reps 1 6 1 2 8 12 2 16 1 4 3

Matrix representation of C22×C7⋊C12 in GL8(𝔽337)

 1 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336
,
 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 336 0 0 1 0 0 0 0 336 0 0 0 1 0 0 0 336 0 0 0 0 1 0 0 336 0 0 0 0 0 1 0 336 0 0 0 0 0 0 1 336
,
 128 0 0 0 0 0 0 0 0 129 0 0 0 0 0 0 0 0 46 0 0 46 22 291 0 0 46 46 22 0 291 0 0 0 68 0 291 46 291 0 0 0 0 46 291 46 0 22 0 0 0 46 0 68 291 291 0 0 46 68 291 0 0 291

G:=sub<GL(8,GF(337))| [1,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,336,336,336,336,336,336],[128,0,0,0,0,0,0,0,0,129,0,0,0,0,0,0,0,0,46,46,68,0,0,46,0,0,0,46,0,46,46,68,0,0,0,22,291,291,0,291,0,0,46,0,46,46,68,0,0,0,22,291,291,0,291,0,0,0,291,0,0,22,291,291] >;

C22×C7⋊C12 in GAP, Magma, Sage, TeX

C_2^2\times C_7\rtimes C_{12}
% in TeX

G:=Group("C2^2xC7:C12");
// GroupNames label

G:=SmallGroup(336,129);
// by ID

G=gap.SmallGroup(336,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,144,10373,887]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^7=d^12=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^5>;
// generators/relations

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