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## G = C2×D7×M4(2)  order 448 = 26·7

### Direct product of C2, D7 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×D7×M4(2)
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — D7×C22×C4 — C2×D7×M4(2)
 Lower central C7 — C14 — C2×D7×M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C2×D7×M4(2)
G = < a,b,c,d,e | a2=b7=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 1124 in 298 conjugacy classes, 159 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, D7, D7, C14, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×D7, C22×C14, C22×M4(2), C8×D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×C4×D7, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, D7×C2×C8, C2×C8⋊D7, D7×M4(2), C2×C4.Dic7, C14×M4(2), D7×C22×C4, C2×D7×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, C24, D14, C2×M4(2), C23×C4, C4×D7, C22×D7, C22×M4(2), C2×C4×D7, C23×D7, D7×M4(2), D7×C22×C4, C2×D7×M4(2)

Smallest permutation representation of C2×D7×M4(2)
On 112 points
Generators in S112
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(33 109)(34 110)(35 111)(36 112)(37 105)(38 106)(39 107)(40 108)(41 96)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 81)(73 100)(74 101)(75 102)(76 103)(77 104)(78 97)(79 98)(80 99)
(1 102 14 89 107 68 50)(2 103 15 90 108 69 51)(3 104 16 91 109 70 52)(4 97 9 92 110 71 53)(5 98 10 93 111 72 54)(6 99 11 94 112 65 55)(7 100 12 95 105 66 56)(8 101 13 96 106 67 49)(17 84 25 76 59 43 40)(18 85 26 77 60 44 33)(19 86 27 78 61 45 34)(20 87 28 79 62 46 35)(21 88 29 80 63 47 36)(22 81 30 73 64 48 37)(23 82 31 74 57 41 38)(24 83 32 75 58 42 39)
(1 87)(2 88)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 97)(24 98)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(41 92)(42 93)(43 94)(44 95)(45 96)(46 89)(47 90)(48 91)(57 110)(58 111)(59 112)(60 105)(61 106)(62 107)(63 108)(64 109)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(97 101)(99 103)(106 110)(108 112)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 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 8 ··· 8 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 7 7 7 7 14 14 1 1 1 1 2 2 7 7 7 7 14 14 2 2 2 2 ··· 2 14 ··· 14 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 D7 M4(2) D14 D14 D14 C4×D7 C4×D7 D7×M4(2) kernel C2×D7×M4(2) D7×C2×C8 C2×C8⋊D7 D7×M4(2) C2×C4.Dic7 C14×M4(2) D7×C22×C4 C2×C4×D7 C22×Dic7 C23×D7 C2×M4(2) D14 C2×C8 M4(2) C22×C4 C2×C4 C23 C2 # reps 1 2 2 8 1 1 1 12 2 2 3 8 6 12 3 18 6 12

Matrix representation of C2×D7×M4(2) in GL5(𝔽113)

 112 0 0 0 0 0 112 0 0 0 0 0 112 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 88 2 0 0 0 112 104
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 88 59 0 0 0 112 25
,
 1 0 0 0 0 0 0 112 0 0 0 98 0 0 0 0 0 0 98 0 0 0 0 0 98
,
 112 0 0 0 0 0 1 0 0 0 0 0 112 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,112,0,0,0,2,104],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,112,0,0,0,59,25],[1,0,0,0,0,0,0,98,0,0,0,112,0,0,0,0,0,0,98,0,0,0,0,0,98],[112,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2×D7×M4(2) in GAP, Magma, Sage, TeX

C_2\times D_7\times M_4(2)
% in TeX

G:=Group("C2xD7xM4(2)");
// GroupNames label

G:=SmallGroup(448,1196);
// by ID

G=gap.SmallGroup(448,1196);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,297,80,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^8=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^5>;
// generators/relations

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