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## G = C2×D4⋊2Dic7order 448 = 26·7

### Direct product of C2 and D4⋊2Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D4⋊2Dic7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C2×C4.Dic7 — C2×D4⋊2Dic7
 Lower central C7 — C14 — C28 — C2×D4⋊2Dic7
 Upper central C1 — C2×C4 — C22×C4 — C2×C4○D4

Generators and relations for C2×D42Dic7
G = < a,b,c,d,e | a2=b4=d14=1, c2=b2, e2=d7, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 532 in 170 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C42, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C2×C4≀C2, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C4×Dic7, C22×Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, D42Dic7, C2×C4.Dic7, C2×C4×Dic7, C14×C4○D4, C2×D42Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C4≀C2, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C2×C4≀C2, C23.D7, C22×Dic7, C2×C7⋊D4, D42Dic7, C2×C23.D7, C2×D42Dic7

Smallest permutation representation of C2×D42Dic7
On 112 points
Generators in S112
(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 42)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(22 53)(23 54)(24 55)(25 56)(26 50)(27 51)(28 52)(57 111)(58 112)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 85)
(1 50 39 30)(2 51 40 31)(3 52 41 32)(4 53 42 33)(5 54 36 34)(6 55 37 35)(7 56 38 29)(8 15 43 22)(9 16 44 23)(10 17 45 24)(11 18 46 25)(12 19 47 26)(13 20 48 27)(14 21 49 28)(57 86 64 93)(58 87 65 94)(59 88 66 95)(60 89 67 96)(61 90 68 97)(62 91 69 98)(63 92 70 85)(71 104 78 111)(72 105 79 112)(73 106 80 99)(74 107 81 100)(75 108 82 101)(76 109 83 102)(77 110 84 103)
(1 92 39 85)(2 86 40 93)(3 94 41 87)(4 88 42 95)(5 96 36 89)(6 90 37 97)(7 98 38 91)(8 80 43 73)(9 74 44 81)(10 82 45 75)(11 76 46 83)(12 84 47 77)(13 78 48 71)(14 72 49 79)(15 106 22 99)(16 100 23 107)(17 108 24 101)(18 102 25 109)(19 110 26 103)(20 104 27 111)(21 112 28 105)(29 62 56 69)(30 70 50 63)(31 64 51 57)(32 58 52 65)(33 66 53 59)(34 60 54 67)(35 68 55 61)
(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 11)(2 10)(3 9)(4 8)(5 14)(6 13)(7 12)(15 53)(16 52)(17 51)(18 50)(19 56)(20 55)(21 54)(22 33)(23 32)(24 31)(25 30)(26 29)(27 35)(28 34)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(57 75 64 82)(58 74 65 81)(59 73 66 80)(60 72 67 79)(61 71 68 78)(62 84 69 77)(63 83 70 76)(85 109 92 102)(86 108 93 101)(87 107 94 100)(88 106 95 99)(89 105 96 112)(90 104 97 111)(91 103 98 110)

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 14 ··· 14 2 2 2 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D7 D14 Dic7 Dic7 Dic7 D14 C4≀C2 C7⋊D4 C7⋊D4 D4⋊2Dic7 kernel C2×D4⋊2Dic7 D4⋊2Dic7 C2×C4.Dic7 C2×C4×Dic7 C14×C4○D4 D4×C14 Q8×C14 C7×C4○D4 C2×C28 C22×C14 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C14 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 3 1 3 3 3 3 6 6 8 18 6 12

Matrix representation of C2×D42Dic7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 98 0 0 0 38 15 0 0 0 0 112 0 0 0 0 112
,
 20 105 0 0 36 93 0 0 0 0 79 108 0 0 5 34
,
 1 0 0 0 5 112 0 0 0 0 9 112 0 0 1 0
,
 1 0 0 0 40 98 0 0 0 0 112 0 0 0 104 1
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[98,38,0,0,0,15,0,0,0,0,112,0,0,0,0,112],[20,36,0,0,105,93,0,0,0,0,79,5,0,0,108,34],[1,5,0,0,0,112,0,0,0,0,9,1,0,0,112,0],[1,40,0,0,0,98,0,0,0,0,112,104,0,0,0,1] >;

C2×D42Dic7 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2{\rm Dic}_7
% in TeX

G:=Group("C2xD4:2Dic7");
// GroupNames label

G:=SmallGroup(448,769);
// by ID

G=gap.SmallGroup(448,769);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,136,1684,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^14=1,c^2=b^2,e^2=d^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

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