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## G = D7×C24order 224 = 25·7

### Direct product of C24 and D7

Aliases: D7×C24, C7⋊C25, C14⋊C24, (C2×C14)⋊4C23, (C23×C14)⋊5C2, (C22×C14)⋊8C22, SmallGroup(224,196)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C24
 Chief series C1 — C7 — D7 — D14 — C22×D7 — C23×D7 — D7×C24
 Lower central C7 — D7×C24
 Upper central C1 — C24

Generators and relations for D7×C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=e7=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2590 in 748 conjugacy classes, 441 normal (5 characteristic)
C1, C2, C2, C22, C22, C7, C23, C23, D7, C14, C24, C24, D14, C2×C14, C25, C22×D7, C22×C14, C23×D7, C23×C14, D7×C24
Quotients: C1, C2, C22, C23, D7, C24, D14, C25, C22×D7, C23×D7, D7×C24

Smallest permutation representation of D7×C24
On 112 points
Generators in S112
(1 111)(2 112)(3 106)(4 107)(5 108)(6 109)(7 110)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(1 55)(2 56)(3 50)(4 51)(5 52)(6 53)(7 54)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(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 82)(2 81)(3 80)(4 79)(5 78)(6 84)(7 83)(8 73)(9 72)(10 71)(11 77)(12 76)(13 75)(14 74)(15 66)(16 65)(17 64)(18 70)(19 69)(20 68)(21 67)(22 59)(23 58)(24 57)(25 63)(26 62)(27 61)(28 60)(29 108)(30 107)(31 106)(32 112)(33 111)(34 110)(35 109)(36 101)(37 100)(38 99)(39 105)(40 104)(41 103)(42 102)(43 94)(44 93)(45 92)(46 98)(47 97)(48 96)(49 95)(50 87)(51 86)(52 85)(53 91)(54 90)(55 89)(56 88)

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

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

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

D7×C24 is a maximal subgroup of   C23.44D28
D7×C24 is a maximal quotient of   C14.C25  D14.C24  D28.39C23

80 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 7A 7B 7C 14A ··· 14AS order 1 2 ··· 2 2 ··· 2 7 7 7 14 ··· 14 size 1 1 ··· 1 7 ··· 7 2 2 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D7 D14 kernel D7×C24 C23×D7 C23×C14 C24 C23 # reps 1 30 1 3 45

Matrix representation of D7×C24 in GL5(𝔽29)

 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 28 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 28 18
,
 28 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 28 0

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,1,18],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,28,0] >;

D7×C24 in GAP, Magma, Sage, TeX

D_7\times C_2^4
% in TeX

G:=Group("D7xC2^4");
// GroupNames label

G:=SmallGroup(224,196);
// by ID

G=gap.SmallGroup(224,196);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,6917]);
// Polycyclic

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

׿
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𝔽