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## G = C22⋊Dic14order 224 = 25·7

### The semidirect product of C22 and Dic14 acting via Dic14/Dic7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C22⋊Dic14
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C22⋊Dic14
 Lower central C7 — C2×C14 — C22⋊Dic14
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22⋊Dic14
G = < a,b,c,d | a2=b2=c28=1, d2=c14, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 262 in 74 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C22⋊Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C22×C14, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C22⋊C4, C2×Dic14, C22×Dic7, C22⋊Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, Dic14, C22×D7, C2×Dic14, D4×D7, D42D7, C22⋊Dic14

Smallest permutation representation of C22⋊Dic14
On 112 points
Generators in S112
(1 15)(2 83)(3 17)(4 57)(5 19)(6 59)(7 21)(8 61)(9 23)(10 63)(11 25)(12 65)(13 27)(14 67)(16 69)(18 71)(20 73)(22 75)(24 77)(26 79)(28 81)(29 43)(30 102)(31 45)(32 104)(33 47)(34 106)(35 49)(36 108)(37 51)(38 110)(39 53)(40 112)(41 55)(42 86)(44 88)(46 90)(48 92)(50 94)(52 96)(54 98)(56 100)(58 72)(60 74)(62 76)(64 78)(66 80)(68 82)(70 84)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 85)(56 86)
(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 87 15 101)(2 86 16 100)(3 85 17 99)(4 112 18 98)(5 111 19 97)(6 110 20 96)(7 109 21 95)(8 108 22 94)(9 107 23 93)(10 106 24 92)(11 105 25 91)(12 104 26 90)(13 103 27 89)(14 102 28 88)(29 82 43 68)(30 81 44 67)(31 80 45 66)(32 79 46 65)(33 78 47 64)(34 77 48 63)(35 76 49 62)(36 75 50 61)(37 74 51 60)(38 73 52 59)(39 72 53 58)(40 71 54 57)(41 70 55 84)(42 69 56 83)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 4 4 14 14 14 14 28 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D7 C4○D4 D14 D14 Dic14 D4×D7 D4⋊2D7 kernel C22⋊Dic14 Dic7⋊C4 C4⋊Dic7 C23.D7 C7×C22⋊C4 C2×Dic14 C22×Dic7 Dic7 C2×C14 C22⋊C4 C14 C2×C4 C23 C22 C2 C2 # reps 1 2 1 1 1 1 1 2 2 3 2 6 3 12 3 3

Matrix representation of C22⋊Dic14 in GL6(𝔽29)

 1 0 0 0 0 0 0 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
,
 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 1 0 0 0 0 0 0 1
,
 3 28 0 0 0 0 2 19 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 23 0
,
 3 28 0 0 0 0 8 26 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 20 3 0 0 0 0 21 9

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,2,0,0,0,0,28,19,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,23,0,0,0,0,5,0],[3,8,0,0,0,0,28,26,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,20,21,0,0,0,0,3,9] >;

C22⋊Dic14 in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm Dic}_{14}
% in TeX

G:=Group("C2^2:Dic14");
// GroupNames label

G:=SmallGroup(224,73);
// by ID

G=gap.SmallGroup(224,73);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,218,188,50,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^28=1,d^2=c^14,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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