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## G = C22×D4×D7order 448 = 26·7

### Direct product of C22, D4 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C22×D4×D7
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — D7×C24 — C22×D4×D7
 Lower central C7 — C14 — C22×D4×D7
 Upper central C1 — C23 — C22×D4

Generators and relations for C22×D4×D7
G = < a,b,c,d,e,f | a2=b2=c4=d2=e7=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 6548 in 1362 conjugacy classes, 503 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, C23, C23, C23, D7, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C23×C4, C22×D4, C22×D4, C25, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, D4×C23, C2×C4×D7, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23×D7, C23×D7, C23×C14, D7×C22×C4, C22×D28, C2×D4×D7, C22×C7⋊D4, D4×C2×C14, D7×C24, C22×D4×D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C25, C22×D7, D4×C23, D4×D7, C23×D7, C2×D4×D7, D7×C24, C22×D4×D7

Smallest permutation representation of C22×D4×D7
On 112 points
Generators in S112
(1 69)(2 70)(3 64)(4 65)(5 66)(6 67)(7 68)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 76 13 83)(2 77 14 84)(3 71 8 78)(4 72 9 79)(5 73 10 80)(6 74 11 81)(7 75 12 82)(15 64 22 57)(16 65 23 58)(17 66 24 59)(18 67 25 60)(19 68 26 61)(20 69 27 62)(21 70 28 63)(29 99 36 106)(30 100 37 107)(31 101 38 108)(32 102 39 109)(33 103 40 110)(34 104 41 111)(35 105 42 112)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)
(1 62)(2 63)(3 57)(4 58)(5 59)(6 60)(7 61)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(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 89)(2 88)(3 87)(4 86)(5 85)(6 91)(7 90)(8 94)(9 93)(10 92)(11 98)(12 97)(13 96)(14 95)(15 101)(16 100)(17 99)(18 105)(19 104)(20 103)(21 102)(22 108)(23 107)(24 106)(25 112)(26 111)(27 110)(28 109)(29 59)(30 58)(31 57)(32 63)(33 62)(34 61)(35 60)(36 66)(37 65)(38 64)(39 70)(40 69)(41 68)(42 67)(43 73)(44 72)(45 71)(46 77)(47 76)(48 75)(49 74)(50 80)(51 79)(52 78)(53 84)(54 83)(55 82)(56 81)

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

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

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

100 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 2P ··· 2W 2X ··· 2AE 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 ··· 2 7 ··· 7 14 ··· 14 2 2 2 2 14 14 14 14 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D7 D14 D14 D14 D4×D7 kernel C22×D4×D7 D7×C22×C4 C22×D28 C2×D4×D7 C22×C7⋊D4 D4×C2×C14 D7×C24 C22×D7 C22×D4 C22×C4 C2×D4 C24 C22 # reps 1 1 1 24 2 1 2 8 3 3 36 6 12

Matrix representation of C22×D4×D7 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 13 0 0 0 0 11 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 11 1
,
 11 1 0 0 0 0 13 25 0 0 0 0 0 0 0 1 0 0 0 0 28 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 1 0 0 0 0 14 25 0 0 0 0 0 0 0 28 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,11,0,0,0,0,13,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,11,0,0,0,0,0,1],[11,13,0,0,0,0,1,25,0,0,0,0,0,0,0,28,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,14,0,0,0,0,1,25,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22×D4×D7 in GAP, Magma, Sage, TeX

C_2^2\times D_4\times D_7
% in TeX

G:=Group("C2^2xD4xD7");
// GroupNames label

G:=SmallGroup(448,1369);
// by ID

G=gap.SmallGroup(448,1369);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,235,18822]);
// Polycyclic

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

׿
×
𝔽