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G = C42⋊2Dic7order 448 = 26·7

2nd semidirect product of C42 and Dic7 acting via Dic7/C7=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C42⋊2Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — D4×C14 — C23⋊Dic7 — C42⋊2Dic7
 Lower central C7 — C14 — C2×C14 — C2×C28 — C42⋊2Dic7
 Upper central C1 — C2 — C22 — C2×D4 — C4.4D4

Generators and relations for C422Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 364 in 70 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C4.4D4, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C423C4, C23.D7, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C23⋊Dic7, C7×C4.4D4, C422Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C423C4, C23.D7, C23⋊Dic7, C422Dic7

Smallest permutation representation of C422Dic7
On 112 points
Generators in S112
(1 98)(3 86)(5 88)(7 90)(9 92)(11 94)(13 96)(16 72)(18 74)(20 76)(22 78)(24 80)(26 82)(28 84)(29 53 110 57)(30 54 111 58)(31 55 112 59)(32 56 99 60)(33 43 100 61)(34 44 101 62)(35 45 102 63)(36 46 103 64)(37 47 104 65)(38 48 105 66)(39 49 106 67)(40 50 107 68)(41 51 108 69)(42 52 109 70)
(1 76 98 20)(2 21 85 77)(3 78 86 22)(4 23 87 79)(5 80 88 24)(6 25 89 81)(7 82 90 26)(8 27 91 83)(9 84 92 28)(10 15 93 71)(11 72 94 16)(12 17 95 73)(13 74 96 18)(14 19 97 75)(29 53 110 57)(30 58 111 54)(31 55 112 59)(32 60 99 56)(33 43 100 61)(34 62 101 44)(35 45 102 63)(36 64 103 46)(37 47 104 65)(38 66 105 48)(39 49 106 67)(40 68 107 50)(41 51 108 69)(42 70 109 52)
(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 48 8 55)(2 47 9 54)(3 46 10 53)(4 45 11 52)(5 44 12 51)(6 43 13 50)(7 56 14 49)(15 29 22 36)(16 42 23 35)(17 41 24 34)(18 40 25 33)(19 39 26 32)(20 38 27 31)(21 37 28 30)(57 86 64 93)(58 85 65 92)(59 98 66 91)(60 97 67 90)(61 96 68 89)(62 95 69 88)(63 94 70 87)(71 110 78 103)(72 109 79 102)(73 108 80 101)(74 107 81 100)(75 106 82 99)(76 105 83 112)(77 104 84 111)

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

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

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

55 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28R 28S ··· 28X order 1 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 4 4 4 4 4 8 56 56 56 56 2 2 2 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 D14 Dic7 C7⋊D4 C23⋊C4 C42⋊3C4 C23⋊Dic7 C42⋊2Dic7 kernel C42⋊2Dic7 C23⋊Dic7 C7×C4.4D4 C4×C28 Q8×C14 C22×C14 C4.4D4 C42 C2×D4 C2×Q8 C23 C14 C7 C2 C1 # reps 1 2 1 2 2 2 3 3 3 3 12 1 2 6 12

Matrix representation of C422Dic7 in GL4(𝔽29) generated by

 28 0 0 0 0 1 0 0 0 0 17 0 0 0 0 17
,
 17 0 0 0 0 12 0 0 0 0 17 0 0 0 0 12
,
 0 20 0 0 20 0 0 0 0 0 0 16 0 0 16 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,17,0,0,0,0,17],[17,0,0,0,0,12,0,0,0,0,17,0,0,0,0,12],[0,20,0,0,20,0,0,0,0,0,0,16,0,0,16,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;

C422Dic7 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2{\rm Dic}_7
% in TeX

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

G:=SmallGroup(448,98);
// by ID

G=gap.SmallGroup(448,98);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,219,1571,570,297,136,1684,18822]);
// Polycyclic

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

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