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G = C42:2D7order 224 = 25·7

2nd semidirect product of C42 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:2D7, (C4xC28):1C2, Dic7:C4:1C2, D14:C4.1C2, (C2xC4).63D14, C7:1(C42:2C2), C14.6(C4oD4), C2.8(C4oD28), (C2xC14).17C23, (C2xC28).75C22, (C2xDic7).4C22, (C22xD7).3C22, C22.38(C22xD7), SmallGroup(224,71)

Series: Derived Chief Lower central Upper central

C1C2xC14 — C42:2D7
C1C7C14C2xC14C22xD7D14:C4 — C42:2D7
C7C2xC14 — C42:2D7
C1C22C42

Generators and relations for C42:2D7
 G = < a,b,c,d | a4=b4=c7=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >

Subgroups: 246 in 60 conjugacy classes, 29 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2xC4, C2xC4, C23, D7, C14, C42, C22:C4, C4:C4, Dic7, C28, D14, C2xC14, C42:2C2, C2xDic7, C2xC28, C22xD7, Dic7:C4, D14:C4, C4xC28, C42:2D7
Quotients: C1, C2, C22, C23, D7, C4oD4, D14, C42:2C2, C22xD7, C4oD28, C42:2D7

Smallest permutation representation of C42:2D7
On 112 points
Generators in S112
(1 76 20 62)(2 77 21 63)(3 71 15 57)(4 72 16 58)(5 73 17 59)(6 74 18 60)(7 75 19 61)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)
(1 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 106 78 99)(72 107 79 100)(73 108 80 101)(74 109 81 102)(75 110 82 103)(76 111 83 104)(77 112 84 105)
(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)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 101)(86 100)(87 99)(88 105)(89 104)(90 103)(91 102)(92 108)(93 107)(94 106)(95 112)(96 111)(97 110)(98 109)

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

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

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

C42:2D7 is a maximal subgroup of
C42.277D14  C42:10D14  C42.95D14  C42.96D14  C42.98D14  C42.104D14  C42:16D14  C42:17D14  C42.118D14  C42.122D14  C42.132D14  C42.133D14  C42.134D14  C42.137D14  C42:20D14  C42.150D14  C42.154D14  D7xC42:2C2  C42.189D14  C42:25D14  C42.165D14  C42:28D14  C42.180D14
C42:2D7 is a maximal quotient of
(C2xDic7).Q8  (C22xC4).D14  (C22xD7).9D4  (C22xD7).Q8  (C2xC42).D7  C42:5Dic7  (C2xC42):D7

62 conjugacy classes

class 1 2A2B2C2D4A···4F4G4H4I7A7B7C14A···14I28A···28AJ
order122224···444477714···1428···28
size1111282···22828282222···22···2

62 irreducible representations

dim11112222
type++++++
imageC1C2C2C2D7C4oD4D14C4oD28
kernelC42:2D7Dic7:C4D14:C4C4xC28C42C14C2xC4C2
# reps133136936

Matrix representation of C42:2D7 in GL4(F29) generated by

91400
152000
00275
00282
,
212300
6800
00170
00017
,
0100
28700
00261
002321
,
0100
1000
002128
0058
G:=sub<GL(4,GF(29))| [9,15,0,0,14,20,0,0,0,0,27,28,0,0,5,2],[21,6,0,0,23,8,0,0,0,0,17,0,0,0,0,17],[0,28,0,0,1,7,0,0,0,0,26,23,0,0,1,21],[0,1,0,0,1,0,0,0,0,0,21,5,0,0,28,8] >;

C42:2D7 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2D_7
% in TeX

G:=Group("C4^2:2D7");
// GroupNames label

G:=SmallGroup(224,71);
// by ID

G=gap.SmallGroup(224,71);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,55,506,86,6917]);
// Polycyclic

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

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