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## G = (C2×Q8).D14order 448 = 26·7

### 2nd non-split extension by C2×Q8 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (C2×Q8).D14
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — Q8×C14 — Dic7⋊Q8 — (C2×Q8).D14
 Lower central C7 — C14 — C2×C14 — C2×C28 — (C2×Q8).D14
 Upper central C1 — C2 — C22 — C2×Q8 — C4.10D4

Generators and relations for (C2×Q8).D14
G = < a,b,c,d | a2=b28=1, c4=d2=b14, ab=ba, cac-1=ab14, ad=da, cbc-1=dbd-1=ab13, dcd-1=b21c3 >

Subgroups: 300 in 60 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, M4(2), C2×Q8, C2×Q8, Dic7, C28, C2×C14, C4.10D4, C4.10D4, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C7×Q8, C42.3C4, C4.Dic7, C4×Dic7, Dic7⋊C4, C7×M4(2), C2×Dic14, Q8×C14, C28.10D4, C7×C4.10D4, Dic7⋊Q8, (C2×Q8).D14
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C42.3C4, D14⋊C4, C23.1D14, (C2×Q8).D14

Smallest permutation representation of (C2×Q8).D14
On 112 points
Generators in S112
(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)
(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 55 107 76 15 41 93 62)(2 40 94 75 16 54 108 61)(3 53 109 74 17 39 95 60)(4 38 96 73 18 52 110 59)(5 51 111 72 19 37 97 58)(6 36 98 71 20 50 112 57)(7 49 85 70 21 35 99 84)(8 34 100 69 22 48 86 83)(9 47 87 68 23 33 101 82)(10 32 102 67 24 46 88 81)(11 45 89 66 25 31 103 80)(12 30 104 65 26 44 90 79)(13 43 91 64 27 29 105 78)(14 56 106 63 28 42 92 77)
(1 22 15 8)(2 7 16 21)(3 20 17 6)(4 5 18 19)(9 14 23 28)(10 27 24 13)(11 12 25 26)(29 60 43 74)(30 59 44 73)(31 58 45 72)(32 57 46 71)(33 84 47 70)(34 83 48 69)(35 82 49 68)(36 81 50 67)(37 80 51 66)(38 79 52 65)(39 78 53 64)(40 77 54 63)(41 76 55 62)(42 75 56 61)(85 94 99 108)(86 107 100 93)(87 92 101 106)(88 105 102 91)(89 90 103 104)(95 112 109 98)(96 97 110 111)

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

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

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

46 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 4 4 4 28 28 56 2 2 2 8 8 56 56 2 2 2 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + - - image C1 C2 C2 C2 C4 C4 D4 D7 D14 C4×D7 D28 C7⋊D4 C23⋊C4 C42.3C4 C23.1D14 (C2×Q8).D14 kernel (C2×Q8).D14 C28.10D4 C7×C4.10D4 Dic7⋊Q8 C4×Dic7 C2×Dic14 C2×C28 C4.10D4 C2×Q8 C2×C4 C2×C4 C2×C4 C14 C7 C2 C1 # reps 1 1 1 1 2 2 2 3 3 6 6 6 1 2 6 3

Matrix representation of (C2×Q8).D14 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 1 89 0 0 0 0 24 103 0 0 0 0 0 0 10 99 0 0 0 0 96 103 0 0 0 0 0 0 10 99 0 0 0 0 96 103
,
 17 8 0 0 0 0 77 96 0 0 0 0 0 0 0 0 89 79 0 0 0 0 7 24 0 0 112 0 0 0 0 0 0 112 0 0
,
 112 0 0 0 0 0 89 1 0 0 0 0 0 0 10 99 0 0 0 0 96 103 0 0 0 0 0 0 89 79 0 0 0 0 7 24

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,24,0,0,0,0,89,103,0,0,0,0,0,0,10,96,0,0,0,0,99,103,0,0,0,0,0,0,10,96,0,0,0,0,99,103],[17,77,0,0,0,0,8,96,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,89,7,0,0,0,0,79,24,0,0],[112,89,0,0,0,0,0,1,0,0,0,0,0,0,10,96,0,0,0,0,99,103,0,0,0,0,0,0,89,7,0,0,0,0,79,24] >;

(C2×Q8).D14 in GAP, Magma, Sage, TeX

(C_2\times Q_8).D_{14}
% in TeX

G:=Group("(C2xQ8).D14");
// GroupNames label

G:=SmallGroup(448,35);
// by ID

G=gap.SmallGroup(448,35);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,184,1123,794,297,136,851,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=1,c^4=d^2=b^14,a*b=b*a,c*a*c^-1=a*b^14,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^13,d*c*d^-1=b^21*c^3>;
// generators/relations

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