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G = D7xC42:2C2order 448 = 26·7

Direct product of D7 and C42:2C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D7xC42:2C2, C42:30D14, C4:C4:32D14, (C4xC28):31C22, (D7xC42):19C2, C42:2D7:13C2, (C2xC28).94C23, C4:Dic7:43C22, C22:C4.76D14, D14.42(C4oD4), (C2xC14).247C24, Dic7:C4:31C22, D14:C4.44C22, (C4xDic7):80C22, C23.53(C22xD7), C23.D14:44C2, (C22xC14).61C23, (C23xD7).67C22, C22.268(C23xD7), C23.D7.63C22, (C2xDic7).128C23, (C22xD7).258C23, (D7xC4:C4):40C2, C7:4(C2xC42:2C2), C2.94(D7xC4oD4), C4:C4:D7:40C2, (C7xC4:C4):31C22, (D7xC22:C4).3C2, (C7xC42:2C2):2C2, C14.205(C2xC4oD4), (C2xC4xD7).299C22, (C2xC4).84(C22xD7), (C7xC22:C4).72C22, SmallGroup(448,1156)

Series: Derived Chief Lower central Upper central

C1C2xC14 — D7xC42:2C2
C1C7C14C2xC14C22xD7C23xD7D7xC22:C4 — D7xC42:2C2
C7C2xC14 — D7xC42:2C2
C1C22C42:2C2

Generators and relations for D7xC42:2C2
 G = < a,b,c,d,e | a7=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, ede=c2d-1 >

Subgroups: 1164 in 246 conjugacy classes, 101 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2xC4, C2xC4, C23, C23, D7, D7, C14, C14, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C24, Dic7, C28, D14, D14, C2xC14, C2xC14, C2xC42, C2xC22:C4, C2xC4:C4, C42:2C2, C42:2C2, C4xD7, C2xDic7, C2xC28, C22xD7, C22xD7, C22xC14, C2xC42:2C2, C4xDic7, Dic7:C4, C4:Dic7, D14:C4, C23.D7, C4xC28, C7xC22:C4, C7xC4:C4, C2xC4xD7, C23xD7, D7xC42, C42:2D7, C23.D14, D7xC22:C4, D7xC4:C4, C4:C4:D7, C7xC42:2C2, D7xC42:2C2
Quotients: C1, C2, C22, C23, D7, C4oD4, C24, D14, C42:2C2, C2xC4oD4, C22xD7, C2xC42:2C2, C23xD7, D7xC4oD4, D7xC42:2C2

Smallest permutation representation of D7xC42:2C2
On 112 points
Generators in S112
(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 19)(2 18)(3 17)(4 16)(5 15)(6 21)(7 20)(8 24)(9 23)(10 22)(11 28)(12 27)(13 26)(14 25)(29 45)(30 44)(31 43)(32 49)(33 48)(34 47)(35 46)(36 52)(37 51)(38 50)(39 56)(40 55)(41 54)(42 53)(57 73)(58 72)(59 71)(60 77)(61 76)(62 75)(63 74)(64 80)(65 79)(66 78)(67 84)(68 83)(69 82)(70 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)
(1 97 13 90)(2 98 14 91)(3 92 8 85)(4 93 9 86)(5 94 10 87)(6 95 11 88)(7 96 12 89)(15 106 22 99)(16 107 23 100)(17 108 24 101)(18 109 25 102)(19 110 26 103)(20 111 27 104)(21 112 28 105)(29 71 36 78)(30 72 37 79)(31 73 38 80)(32 74 39 81)(33 75 40 82)(34 76 41 83)(35 77 42 84)(43 57 50 64)(44 58 51 65)(45 59 52 66)(46 60 53 67)(47 61 54 68)(48 62 55 69)(49 63 56 70)
(1 55 27 34)(2 56 28 35)(3 50 22 29)(4 51 23 30)(5 52 24 31)(6 53 25 32)(7 54 26 33)(8 43 15 36)(9 44 16 37)(10 45 17 38)(11 46 18 39)(12 47 19 40)(13 48 20 41)(14 49 21 42)(57 106 78 85)(58 107 79 86)(59 108 80 87)(60 109 81 88)(61 110 82 89)(62 111 83 90)(63 112 84 91)(64 99 71 92)(65 100 72 93)(66 101 73 94)(67 102 74 95)(68 103 75 96)(69 104 76 97)(70 105 77 98)
(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 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)

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

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

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

70 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4F4G4H4I4J···4O4P4Q4R7A7B7C14A···14I14J14K14L28A···28R28S···28AA
order12222222224···44444···444477714···1414141428···2828···28
size111147777282···244414···142828282222···28884···48···8

70 irreducible representations

dim11111111222224
type++++++++++++
imageC1C2C2C2C2C2C2C2D7C4oD4D14D14D14D7xC4oD4
kernelD7xC42:2C2D7xC42C42:2D7C23.D14D7xC22:C4D7xC4:C4C4:C4:D7C7xC42:2C2C42:2C2D14C42C22:C4C4:C4C2
# reps1113333131239918

Matrix representation of D7xC42:2C2 in GL6(F29)

100000
010000
0026100
00271000
000010
000001
,
2800000
0280000
00102800
00121900
0000280
0000028
,
1200000
0120000
0028000
0002800
00001227
00002817
,
010000
100000
001000
000100
0000124
00001228
,
100000
0280000
001000
000100
000010
00001228

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,27,0,0,0,0,1,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,12,0,0,0,0,28,19,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,28,0,0,0,0,27,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,24,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,28] >;

D7xC42:2C2 in GAP, Magma, Sage, TeX

D_7\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("D7xC4^2:2C2");
// GroupNames label

G:=SmallGroup(448,1156);
// by ID

G=gap.SmallGroup(448,1156);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,100,346,297,136,18822]);
// Polycyclic

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

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