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## G = C2×D4⋊D7order 224 = 25·7

### Direct product of C2 and D4⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D4⋊D7
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C2×D4⋊D7
 Lower central C7 — C14 — C28 — C2×D4⋊D7
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4⋊D7
G = < a,b,c,d,e | a2=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 350 in 76 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, D4, C23, D7, C14, C14, C14, C2×C8, D8, C2×D4, C2×D4, C28, D14, C2×C14, C2×C14, C2×D8, C7⋊C8, D28, D28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C2×C7⋊C8, D4⋊D7, C2×D28, D4×C14, C2×D4⋊D7
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7

Smallest permutation representation of C2×D4⋊D7
On 112 points
Generators in S112
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(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 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(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)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(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 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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14U 28A ··· 28F order 1 2 2 2 2 2 2 2 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 4 28 28 2 2 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D7 D8 D14 D14 C7⋊D4 C7⋊D4 D4⋊D7 kernel C2×D4⋊D7 C2×C7⋊C8 D4⋊D7 C2×D28 D4×C14 C28 C2×C14 C2×D4 C14 C2×C4 D4 C4 C22 C2 # reps 1 1 4 1 1 1 1 3 4 3 6 6 6 6

Matrix representation of C2×D4⋊D7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 0 1 0 0 112 0
,
 91 12 0 0 101 22 0 0 0 0 82 31 0 0 31 31
,
 79 112 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 34 1 0 0 88 79 0 0 0 0 1 0 0 0 0 112
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,0,112,0,0,1,0],[91,101,0,0,12,22,0,0,0,0,82,31,0,0,31,31],[79,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[34,88,0,0,1,79,0,0,0,0,1,0,0,0,0,112] >;

C2×D4⋊D7 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_7
% in TeX

G:=Group("C2xD4:D7");
// GroupNames label

G:=SmallGroup(224,126);
// by ID

G=gap.SmallGroup(224,126);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,579,159,69,6917]);
// Polycyclic

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

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