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## G = C14×C8⋊C22order 448 = 26·7

### Direct product of C14 and C8⋊C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C14×C8⋊C22
 Chief series C1 — C2 — C4 — C28 — C7×D4 — C7×D8 — C7×C8⋊C22 — C14×C8⋊C22
 Lower central C1 — C2 — C4 — C14×C8⋊C22
 Upper central C1 — C2×C14 — C22×C28 — C14×C8⋊C22

Generators and relations for C14×C8⋊C22
G = < a,b,c,d | a14=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C28, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C2×C8⋊C22, C2×C56, C7×M4(2), C7×D8, C7×SD16, C22×C28, C22×C28, D4×C14, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C23×C14, C14×M4(2), C14×D8, C14×SD16, C7×C8⋊C22, D4×C2×C14, C14×C4○D4, C14×C8⋊C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C8⋊C22, C22×D4, C7×D4, C22×C14, C2×C8⋊C22, D4×C14, C23×C14, C7×C8⋊C22, D4×C2×C14, C14×C8⋊C22

Smallest permutation representation of C14×C8⋊C22
On 112 points
Generators in S112
(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)
(1 74 36 15 54 106 63 94)(2 75 37 16 55 107 64 95)(3 76 38 17 56 108 65 96)(4 77 39 18 43 109 66 97)(5 78 40 19 44 110 67 98)(6 79 41 20 45 111 68 85)(7 80 42 21 46 112 69 86)(8 81 29 22 47 99 70 87)(9 82 30 23 48 100 57 88)(10 83 31 24 49 101 58 89)(11 84 32 25 50 102 59 90)(12 71 33 26 51 103 60 91)(13 72 34 27 52 104 61 92)(14 73 35 28 53 105 62 93)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 43)(12 44)(13 45)(14 46)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 98)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)
(15 94)(16 95)(17 96)(18 97)(19 98)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 99)(82 100)(83 101)(84 102)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2K 4A 4B 4C 4D 4E 4F 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AD 14AE ··· 14BN 28A ··· 28X 28Y ··· 28AJ 56A ··· 56X order 1 2 2 2 2 2 2 ··· 2 4 4 4 4 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 4 ··· 4 2 2 2 2 4 4 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 D4 D4 C7×D4 C7×D4 C8⋊C22 C7×C8⋊C22 kernel C14×C8⋊C22 C14×M4(2) C14×D8 C14×SD16 C7×C8⋊C22 D4×C2×C14 C14×C4○D4 C2×C8⋊C22 C2×M4(2) C2×D8 C2×SD16 C8⋊C22 C22×D4 C2×C4○D4 C2×C28 C22×C14 C2×C4 C23 C14 C2 # reps 1 1 2 2 8 1 1 6 6 12 12 48 6 6 3 1 18 6 2 12

Matrix representation of C14×C8⋊C22 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 112 2 0 0 0 0 112 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 1 0 0 0 0 1 0 0 0
,
 112 0 0 0 0 0 112 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 0 112 0 0 0 0 112 0
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[112,112,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,112,0,0],[112,112,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,112,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;

C14×C8⋊C22 in GAP, Magma, Sage, TeX

C_{14}\times C_8\rtimes C_2^2
% in TeX

G:=Group("C14xC8:C2^2");
// GroupNames label

G:=SmallGroup(448,1356);
// by ID

G=gap.SmallGroup(448,1356);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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