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G = C2×D4.D7order 224 = 25·7

Direct product of C2 and D4.D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D4.D7
 Chief series C1 — C7 — C14 — C28 — Dic14 — C2×Dic14 — C2×D4.D7
 Lower central C7 — C14 — C28 — C2×D4.D7
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4.D7
G = < a,b,c,d,e | a2=b4=c2=d7=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 222 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C14, C2×C8, SD16, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2×SD16, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C7×D4, C7×D4, C22×C14, C2×C7⋊C8, D4.D7, C2×Dic14, D4×C14, C2×D4.D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C2×SD16, C7⋊D4, C22×D7, D4.D7, C2×C7⋊D4, C2×D4.D7

Smallest permutation representation of C2×D4.D7
On 112 points
Generators in S112
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(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)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(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 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)
(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 96 8 89)(2 95 9 88)(3 94 10 87)(4 93 11 86)(5 92 12 85)(6 98 13 91)(7 97 14 90)(15 103 22 110)(16 102 23 109)(17 101 24 108)(18 100 25 107)(19 99 26 106)(20 105 27 112)(21 104 28 111)(29 68 36 61)(30 67 37 60)(31 66 38 59)(32 65 39 58)(33 64 40 57)(34 70 41 63)(35 69 42 62)(43 75 50 82)(44 74 51 81)(45 73 52 80)(46 72 53 79)(47 71 54 78)(48 77 55 84)(49 76 56 83)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14U 28A ··· 28F order 1 2 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 4 2 2 28 28 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D4 D7 SD16 D14 D14 C7⋊D4 C7⋊D4 D4.D7 kernel C2×D4.D7 C2×C7⋊C8 D4.D7 C2×Dic14 D4×C14 C28 C2×C14 C2×D4 C14 C2×C4 D4 C4 C22 C2 # reps 1 1 4 1 1 1 1 3 4 3 6 6 6 6

Matrix representation of C2×D4.D7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 0 1 0 0 112 0
,
 112 0 0 0 10 1 0 0 0 0 0 1 0 0 1 0
,
 106 0 0 0 2 16 0 0 0 0 1 0 0 0 0 1
,
 66 81 0 0 69 47 0 0 0 0 100 13 0 0 13 13
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,0,112,0,0,1,0],[112,10,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[106,2,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[66,69,0,0,81,47,0,0,0,0,100,13,0,0,13,13] >;

C2×D4.D7 in GAP, Magma, Sage, TeX

C_2\times D_4.D_7
% in TeX

G:=Group("C2xD4.D7");
// GroupNames label

G:=SmallGroup(224,128);
// by ID

G=gap.SmallGroup(224,128);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,218,579,159,69,6917]);
// Polycyclic

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

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