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## G = C2×C24⋊D7order 448 = 26·7

### Direct product of C2 and C24⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C24⋊D7
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22×C7⋊D4 — C2×C24⋊D7
 Lower central C7 — C2×C14 — C2×C24⋊D7
 Upper central C1 — C23 — C25

Generators and relations for C2×C24⋊D7
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f7=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 2308 in 662 conjugacy classes, 159 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, C23, C23, C23, D7, C14, C14, C22⋊C4, C22×C4, C2×D4, C24, C24, Dic7, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C22×D4, C25, C2×Dic7, C2×Dic7, C7⋊D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C22≀C2, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C23×D7, C23×C14, C23×C14, C2×C23.D7, C24⋊D7, C22×C7⋊D4, C24×C14, C2×C24⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22≀C2, C22×D4, C7⋊D4, C22×D7, C2×C22≀C2, C2×C7⋊D4, C23×D7, C24⋊D7, C22×C7⋊D4, C2×C24⋊D7

Smallest permutation representation of C2×C24⋊D7
On 112 points
Generators in S112
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(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 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(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)
(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(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 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 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)
(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 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(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 89)(2 88)(3 87)(4 86)(5 85)(6 91)(7 90)(8 96)(9 95)(10 94)(11 93)(12 92)(13 98)(14 97)(15 103)(16 102)(17 101)(18 100)(19 99)(20 105)(21 104)(22 110)(23 109)(24 108)(25 107)(26 106)(27 112)(28 111)(29 61)(30 60)(31 59)(32 58)(33 57)(34 63)(35 62)(36 68)(37 67)(38 66)(39 65)(40 64)(41 70)(42 69)(43 75)(44 74)(45 73)(46 72)(47 71)(48 77)(49 76)(50 82)(51 81)(52 80)(53 79)(54 78)(55 84)(56 83)

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

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

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

124 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T 2U 4A ··· 4F 7A 7B 7C 14A ··· 14CO order 1 2 ··· 2 2 ··· 2 2 2 4 ··· 4 7 7 7 14 ··· 14 size 1 1 ··· 1 2 ··· 2 28 28 28 ··· 28 2 2 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 D4 D7 D14 C7⋊D4 kernel C2×C24⋊D7 C2×C23.D7 C24⋊D7 C22×C7⋊D4 C24×C14 C22×C14 C25 C24 C23 # reps 1 3 8 3 1 12 3 21 72

Matrix representation of C2×C24⋊D7 in GL5(𝔽29)

 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 28 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 28
,
 28 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 25 0 0 0 0 0 7 0 0 0 0 0 20 0 0 0 0 0 16
,
 28 0 0 0 0 0 0 7 0 0 0 25 0 0 0 0 0 0 0 13 0 0 0 9 0

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0,20,0,0,0,0,0,16],[28,0,0,0,0,0,0,25,0,0,0,7,0,0,0,0,0,0,0,9,0,0,0,13,0] >;

C2×C24⋊D7 in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes D_7
% in TeX

G:=Group("C2xC2^4:D7");
// GroupNames label

G:=SmallGroup(448,1293);
// by ID

G=gap.SmallGroup(448,1293);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,675,18822]);
// Polycyclic

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

׿
×
𝔽