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## G = C42⋊21D14order 448 = 26·7

### 21st semidirect product of C42 and D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C42⋊21D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — C42⋊21D14
 Lower central C7 — C2×C14 — C42⋊21D14
 Upper central C1 — C22 — C4.4D4

Generators and relations for C4221D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1228 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C22.45C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C4×C28, C7×C22⋊C4, C2×C4×D7, C22×Dic7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, C42⋊D7, C23.D14, D7×C22⋊C4, Dic74D4, D14.D4, C23.18D14, C23⋊D14, D143Q8, C7×C4.4D4, C4221D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, C23×D7, D46D14, D7×C4○D4, C4221D14

Smallest permutation representation of C4221D14
On 112 points
Generators in S112
(1 80 17 35)(2 74 18 29)(3 82 19 37)(4 76 20 31)(5 84 21 39)(6 78 15 33)(7 72 16 41)(8 83 25 38)(9 77 26 32)(10 71 27 40)(11 79 28 34)(12 73 22 42)(13 81 23 36)(14 75 24 30)(43 90 108 97)(44 58 109 65)(45 92 110 85)(46 60 111 67)(47 94 112 87)(48 62 99 69)(49 96 100 89)(50 64 101 57)(51 98 102 91)(52 66 103 59)(53 86 104 93)(54 68 105 61)(55 88 106 95)(56 70 107 63)
(1 57 12 90)(2 65 13 98)(3 59 14 92)(4 67 8 86)(5 61 9 94)(6 69 10 88)(7 63 11 96)(15 62 27 95)(16 70 28 89)(17 64 22 97)(18 58 23 91)(19 66 24 85)(20 60 25 93)(21 68 26 87)(29 109 36 51)(30 45 37 103)(31 111 38 53)(32 47 39 105)(33 99 40 55)(34 49 41 107)(35 101 42 43)(44 81 102 74)(46 83 104 76)(48 71 106 78)(50 73 108 80)(52 75 110 82)(54 77 112 84)(56 79 100 72)
(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 7)(2 6)(3 5)(9 14)(10 13)(11 12)(15 18)(16 17)(19 21)(22 28)(23 27)(24 26)(29 33)(30 32)(34 42)(35 41)(36 40)(37 39)(43 100)(44 99)(45 112)(46 111)(47 110)(48 109)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(71 81)(72 80)(73 79)(74 78)(75 77)(82 84)(85 94)(86 93)(87 92)(88 91)(89 90)(95 98)(96 97)

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

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

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

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28R 28S ··· 28X order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 14 14 14 14 2 2 2 2 4 4 4 14 14 14 14 28 28 28 28 2 2 2 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8

67 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D14 2+ 1+4 D4⋊6D14 D7×C4○D4 kernel C42⋊21D14 C42⋊D7 C23.D14 D7×C22⋊C4 Dic7⋊4D4 D14.D4 C23.18D14 C23⋊D14 D14⋊3Q8 C7×C4.4D4 C4.4D4 D14 C42 C22⋊C4 C2×D4 C2×Q8 C14 C2 C2 # reps 1 2 2 2 2 2 1 1 2 1 3 8 3 12 3 3 1 6 12

Matrix representation of C4221D14 in GL6(𝔽29)

 17 0 0 0 0 0 0 17 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 27 0 0 0 0 0 1
,
 1 28 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 17 0 0 0 0 0 0 17
,
 1 0 0 0 0 0 2 28 0 0 0 0 0 0 10 8 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 28 28
,
 1 0 0 0 0 0 2 28 0 0 0 0 0 0 22 26 0 0 0 0 16 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(29))| [17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,27,1],[1,0,0,0,0,0,28,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,10,12,0,0,0,0,8,1,0,0,0,0,0,0,1,28,0,0,0,0,0,28],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,22,16,0,0,0,0,26,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C4221D14 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{21}D_{14}
% in TeX

G:=Group("C4^2:21D14");
// GroupNames label

G:=SmallGroup(448,1132);
// by ID

G=gap.SmallGroup(448,1132);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,387,100,346,136,18822]);
// Polycyclic

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

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