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## G = C42.Dic7order 448 = 26·7

### 2nd non-split extension by C42 of Dic7 acting via Dic7/C7=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C42.Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — Q8×C14 — C28.10D4 — C42.Dic7
 Lower central C7 — C14 — C2×C14 — C2×C28 — C42.Dic7
 Upper central C1 — C2 — C22 — C2×Q8 — C4.4D4

Generators and relations for C42.Dic7
G = < a,b,c,d | a4=b4=1, c14=b2, d2=b2c7, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c13 >

Subgroups: 236 in 64 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C28, C2×C14, C2×C14, C4.10D4, C4.4D4, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C42.C4, C4.Dic7, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C28.10D4, C7×C4.4D4, C42.Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C42.C4, C23.D7, C23⋊Dic7, C42.Dic7

Smallest permutation representation of C42.Dic7
On 112 points
Generators in S112
(1 15)(3 17)(5 19)(7 21)(9 23)(11 25)(13 27)(29 68 43 82)(30 69 44 83)(31 70 45 84)(32 71 46 57)(33 72 47 58)(34 73 48 59)(35 74 49 60)(36 75 50 61)(37 76 51 62)(38 77 52 63)(39 78 53 64)(40 79 54 65)(41 80 55 66)(42 81 56 67)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(1 87 15 101)(2 102 16 88)(3 89 17 103)(4 104 18 90)(5 91 19 105)(6 106 20 92)(7 93 21 107)(8 108 22 94)(9 95 23 109)(10 110 24 96)(11 97 25 111)(12 112 26 98)(13 99 27 85)(14 86 28 100)(29 82 43 68)(30 69 44 83)(31 84 45 70)(32 71 46 57)(33 58 47 72)(34 73 48 59)(35 60 49 74)(36 75 50 61)(37 62 51 76)(38 77 52 63)(39 64 53 78)(40 79 54 65)(41 66 55 80)(42 81 56 67)
(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 66 22 59 15 80 8 73)(2 79 23 72 16 65 9 58)(3 64 24 57 17 78 10 71)(4 77 25 70 18 63 11 84)(5 62 26 83 19 76 12 69)(6 75 27 68 20 61 13 82)(7 60 28 81 21 74 14 67)(29 92 50 85 43 106 36 99)(30 105 51 98 44 91 37 112)(31 90 52 111 45 104 38 97)(32 103 53 96 46 89 39 110)(33 88 54 109 47 102 40 95)(34 101 55 94 48 87 41 108)(35 86 56 107 49 100 42 93)

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

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

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

55 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4E 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28R 28S ··· 28X order 1 2 2 2 4 ··· 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 8 4 ··· 4 2 2 2 56 56 56 56 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8

55 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + - - + + image C1 C2 C2 C4 C4 D4 D7 Dic7 Dic7 D14 C7⋊D4 C23⋊C4 C42.C4 C23⋊Dic7 C42.Dic7 kernel C42.Dic7 C28.10D4 C7×C4.4D4 C4×C28 D4×C14 C2×C28 C4.4D4 C42 C2×D4 C2×Q8 C2×C4 C14 C7 C2 C1 # reps 1 2 1 2 2 2 3 3 3 3 12 1 2 6 12

Matrix representation of C42.Dic7 in GL4(𝔽113) generated by

 112 0 0 0 94 1 0 0 0 0 98 0 0 0 0 98
,
 98 0 0 0 54 15 0 0 0 0 98 0 0 0 54 15
,
 16 34 0 0 78 97 0 0 0 0 106 84 0 0 93 7
,
 0 0 1 0 0 0 0 1 1 101 0 0 19 112 0 0
G:=sub<GL(4,GF(113))| [112,94,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[98,54,0,0,0,15,0,0,0,0,98,54,0,0,0,15],[16,78,0,0,34,97,0,0,0,0,106,93,0,0,84,7],[0,0,1,19,0,0,101,112,1,0,0,0,0,1,0,0] >;

C42.Dic7 in GAP, Magma, Sage, TeX

C_4^2.{\rm Dic}_7
% in TeX

G:=Group("C4^2.Dic7");
// GroupNames label

G:=SmallGroup(448,99);
// by ID

G=gap.SmallGroup(448,99);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,184,1571,570,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^13>;
// generators/relations

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