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G = C4xD4xD7order 448 = 26·7

Direct product of C4, D4 and D7

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

Aliases: C4xD4xD7, C42:32D14, C4:C4:55D14, (D4xC28):8C2, C28:13(C2xD4), D28:12(C2xC4), (C4xD28):23C2, C28:1(C22xC4), (D7xC42):3C2, (C4xC28):15C22, C22:C4:52D14, (D4xDic7):44C2, Dic7:11(C2xD4), D14.58(C2xD4), D14:4(C22xC4), (C22xC4):36D14, D28:C4:46C2, D14:C4:61C22, (C2xD4).244D14, C14.21(C23xC4), (C2xC14).88C24, C4:Dic7:72C22, Dic7:2(C22xC4), C14.46(C22xD4), D14.35(C4oD4), Dic7:4D4:51C2, (C2xC28).586C23, Dic7:C4:63C22, (C22xC28):35C22, (C4xDic7):78C22, C23.D7:47C22, C22.31(C23xD7), (D4xC14).252C22, (C2xD28).257C22, C23.167(C22xD7), (C22xC14).158C23, (C2xDic7).308C23, (C22xDic7):43C22, (C23xD7).105C22, (C22xD7).254C23, C7:4(C2xC4xD4), C4:1(C2xC4xD7), C2.5(C2xD4xD7), C22:1(C2xC4xD7), (D7xC4:C4):47C2, (C4xD7):7(C2xC4), C7:D4:2(C2xC4), (C2xD4xD7).11C2, C2.4(D7xC4oD4), (C7xD4):11(C2xC4), (C4xC7:D4):39C2, (C2xC4xD7):69C22, (D7xC22xC4):21C2, (C7xC4:C4):55C22, C2.23(D7xC22xC4), (C2xC14):1(C22xC4), (D7xC22:C4):30C2, C14.138(C2xC4oD4), (C22xD7):12(C2xC4), (C7xC22:C4):62C22, (C2xC4).819(C22xD7), (C2xC7:D4).109C22, SmallGroup(448,997)

Series: Derived Chief Lower central Upper central

C1C14 — C4xD4xD7
C1C7C14C2xC14C22xD7C23xD7C2xD4xD7 — C4xD4xD7
C7C14 — C4xD4xD7
C1C2xC4C4xD4

Generators and relations for C4xD4xD7
 G = < a,b,c,d,e | a4=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1940 in 426 conjugacy classes, 169 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, C2xC4, D4, D4, C23, C23, D7, D7, C14, C14, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C24, Dic7, Dic7, C28, C28, D14, D14, C2xC14, C2xC14, C2xC14, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C4xD4, C23xC4, C22xD4, C4xD7, C4xD7, D28, C2xDic7, C2xDic7, C2xDic7, C7:D4, C2xC28, C2xC28, C2xC28, C7xD4, C22xD7, C22xD7, C22xD7, C22xC14, C2xC4xD4, C4xDic7, Dic7:C4, C4:Dic7, D14:C4, C23.D7, C4xC28, C7xC22:C4, C7xC4:C4, C2xC4xD7, C2xC4xD7, C2xC4xD7, C2xD28, D4xD7, C22xDic7, C2xC7:D4, C22xC28, D4xC14, C23xD7, D7xC42, C4xD28, D7xC22:C4, Dic7:4D4, D7xC4:C4, D28:C4, C4xC7:D4, D4xDic7, D4xC28, D7xC22xC4, C2xD4xD7, C4xD4xD7
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22xC4, C2xD4, C4oD4, C24, D14, C4xD4, C23xC4, C22xD4, C2xC4oD4, C4xD7, C22xD7, C2xC4xD4, C2xC4xD7, D4xD7, C23xD7, D7xC22xC4, C2xD4xD7, D7xC4oD4, C4xD4xD7

Smallest permutation representation of C4xD4xD7
On 112 points
Generators in S112
(1 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)
(1 69 13 62)(2 70 14 63)(3 64 8 57)(4 65 9 58)(5 66 10 59)(6 67 11 60)(7 68 12 61)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(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)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(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 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)

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

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

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

100 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4L4M4N4O4P4Q···4X7A7B7C14A···14I14J···14U28A···28L28M···28AJ
order122222222222222244444···444444···477714···1414···1428···2828···28
size1111222277771414141411112···2777714···142222···24···42···24···4

100 irreducible representations

dim111111111111122222222244
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D7C4oD4D14D14D14D14D14C4xD7D4xD7D7xC4oD4
kernelC4xD4xD7D7xC42C4xD28D7xC22:C4Dic7:4D4D7xC4:C4D28:C4C4xC7:D4D4xDic7D4xC28D7xC22xC4C2xD4xD7D4xD7C4xD7C4xD4D14C42C22:C4C4:C4C22xC4C2xD4D4C4C2
# reps11122112112116434363632466

Matrix representation of C4xD4xD7 in GL4(F29) generated by

28000
02800
00120
00012
,
0100
28000
00280
00028
,
28000
0100
00280
00028
,
1000
0100
0001
002818
,
28000
02800
00411
002525
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,12,0,0,0,0,12],[0,28,0,0,1,0,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,18],[28,0,0,0,0,28,0,0,0,0,4,25,0,0,11,25] >;

C4xD4xD7 in GAP, Magma, Sage, TeX

C_4\times D_4\times D_7
% in TeX

G:=Group("C4xD4xD7");
// GroupNames label

G:=SmallGroup(448,997);
// by ID

G=gap.SmallGroup(448,997);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,80,18822]);
// Polycyclic

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

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