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## G = C4×D4×D7order 448 = 26·7

### Direct product of C4, D4 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4×D4×D7
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D4×D7 — C4×D4×D7
 Lower central C7 — C14 — C4×D4×D7
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C4×D4×D7
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, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, D7, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C2×C4×D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×C4×D7, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, D7×C42, C4×D28, D7×C22⋊C4, Dic74D4, D7×C4⋊C4, D28⋊C4, C4×C7⋊D4, D4×Dic7, D4×C28, D7×C22×C4, C2×D4×D7, C4×D4×D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, C24, D14, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×D7, C22×D7, C2×C4×D4, C2×C4×D7, D4×D7, C23×D7, D7×C22×C4, C2×D4×D7, D7×C4○D4, C4×D4×D7

Smallest permutation representation of C4×D4×D7
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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 4Q ··· 4X 7A 7B 7C 14A ··· 14I 14J ··· 14U 28A ··· 28L 28M ··· 28AJ order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 2 2 7 7 7 7 14 14 14 14 1 1 1 1 2 ··· 2 7 7 7 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 D7 C4○D4 D14 D14 D14 D14 D14 C4×D7 D4×D7 D7×C4○D4 kernel C4×D4×D7 D7×C42 C4×D28 D7×C22⋊C4 Dic7⋊4D4 D7×C4⋊C4 D28⋊C4 C4×C7⋊D4 D4×Dic7 D4×C28 D7×C22×C4 C2×D4×D7 D4×D7 C4×D7 C4×D4 D14 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C4 C2 # reps 1 1 1 2 2 1 1 2 1 1 2 1 16 4 3 4 3 6 3 6 3 24 6 6

Matrix representation of C4×D4×D7 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 12 0 0 0 0 12
,
 0 1 0 0 28 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 1 0 0 28 18
,
 28 0 0 0 0 28 0 0 0 0 4 11 0 0 25 25
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] >;

C4×D4×D7 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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