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## G = C14×C23⋊C4order 448 = 26·7

### Direct product of C14 and C23⋊C4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C14×C23⋊C4
G = < a,b,c,d,e | a14=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 434 in 210 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C28, C2×C14, C2×C14, C2×C14, C23⋊C4, C2×C22⋊C4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C23⋊C4, C7×C22⋊C4, C7×C22⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, C23×C14, C7×C23⋊C4, C14×C22⋊C4, D4×C2×C14, C14×C23⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C23⋊C4, C2×C22⋊C4, C2×C28, C7×D4, C22×C14, C2×C23⋊C4, C7×C22⋊C4, C22×C28, D4×C14, C7×C23⋊C4, C14×C22⋊C4, C14×C23⋊C4

Smallest permutation representation of C14×C23⋊C4
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 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(29 66)(30 67)(31 68)(32 69)(33 70)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 110)(44 111)(45 112)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 85)(82 86)(83 87)(84 88)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 99)(13 100)(14 101)(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)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 71)(42 72)(57 96)(58 97)(59 98)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)
(1 41)(2 42)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 57)(43 97)(44 98)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 99)(83 100)(84 101)
(1 21 102 89)(2 22 103 90)(3 23 104 91)(4 24 105 92)(5 25 106 93)(6 26 107 94)(7 27 108 95)(8 28 109 96)(9 15 110 97)(10 16 111 98)(11 17 112 85)(12 18 99 86)(13 19 100 87)(14 20 101 88)(29 66 73 51)(30 67 74 52)(31 68 75 53)(32 69 76 54)(33 70 77 55)(34 57 78 56)(35 58 79 43)(36 59 80 44)(37 60 81 45)(38 61 82 46)(39 62 83 47)(40 63 84 48)(41 64 71 49)(42 65 72 50)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A ··· 4J 7A ··· 7F 14A ··· 14R 14S ··· 14BB 14BC ··· 14BN 28A ··· 28BH order 1 2 2 2 2 ··· 2 2 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 4 ··· 4 1 ··· 1 1 ··· 1 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 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C7 C14 C14 C14 C28 C28 C28 D4 C7×D4 C23⋊C4 C7×C23⋊C4 kernel C14×C23⋊C4 C7×C23⋊C4 C14×C22⋊C4 D4×C2×C14 C22×C28 D4×C14 C23×C14 C2×C23⋊C4 C23⋊C4 C2×C22⋊C4 C22×D4 C22×C4 C2×D4 C24 C22×C14 C23 C14 C2 # reps 1 4 2 1 2 4 2 6 24 12 6 12 24 12 4 24 2 12

Matrix representation of C14×C23⋊C4 in GL6(𝔽29)

 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 27 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 20 17 1 2 0 0 28 0 0 0 0 0 19 1 16 12
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 9 12 28 27 0 0 16 13 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 12 24 0 0 0 0 17 17 0 0 0 0 0 0 9 12 28 27 0 0 0 0 28 0 0 0 28 0 0 0 0 0 12 10 4 20

G:=sub<GL(6,GF(29))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,27,1,0,0,0,0,0,0,0,20,28,19,0,0,0,17,0,1,0,0,28,1,0,16,0,0,0,2,0,12],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,9,16,0,0,1,0,12,13,0,0,0,0,28,0,0,0,0,0,27,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,17,0,0,0,0,24,17,0,0,0,0,0,0,9,0,28,12,0,0,12,0,0,10,0,0,28,28,0,4,0,0,27,0,0,20] >;

C14×C23⋊C4 in GAP, Magma, Sage, TeX

C_{14}\times C_2^3\rtimes C_4
% in TeX

G:=Group("C14xC2^3:C4");
// GroupNames label

G:=SmallGroup(448,817);
// by ID

G=gap.SmallGroup(448,817);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,9804,7068]);
// Polycyclic

G:=Group<a,b,c,d,e|a^14=b^2=c^2=d^2=e^4=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,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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