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## G = C7×C22.45C24order 448 = 26·7

### Direct product of C7 and C22.45C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C22.45C24
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — C7×C22.D4 — C7×C22.45C24
 Lower central C1 — C22 — C7×C22.45C24
 Upper central C1 — C2×C14 — C7×C22.45C24

Generators and relations for C7×C22.45C24
G = < a,b,c,d,e,f,g | a7=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 394 in 248 conjugacy classes, 150 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C422C2, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22×C14, C22.45C24, C4×C28, C4×C28, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C23×C14, C14×C22⋊C4, C7×C42⋊C2, D4×C28, C7×C22≀C2, C7×C22⋊Q8, C7×C22.D4, C7×C22.D4, C7×C4.4D4, C7×C422C2, C7×C22.45C24
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, 2+ 1+4, C22×C14, C22.45C24, C7×C4○D4, C23×C14, C14×C4○D4, C7×2+ 1+4, C7×C22.45C24

Smallest permutation representation of C7×C22.45C24
On 112 points
Generators in S112
(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 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(36 47)(37 48)(38 49)(39 43)(40 44)(41 45)(42 46)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)
(1 78 39 94)(2 79 40 95)(3 80 41 96)(4 81 42 97)(5 82 36 98)(6 83 37 92)(7 84 38 93)(8 62 23 77)(9 63 24 71)(10 57 25 72)(11 58 26 73)(12 59 27 74)(13 60 28 75)(14 61 22 76)(15 66 106 50)(16 67 107 51)(17 68 108 52)(18 69 109 53)(19 70 110 54)(20 64 111 55)(21 65 112 56)(29 100 44 85)(30 101 45 86)(31 102 46 87)(32 103 47 88)(33 104 48 89)(34 105 49 90)(35 99 43 91)
(1 63 35 66)(2 57 29 67)(3 58 30 68)(4 59 31 69)(5 60 32 70)(6 61 33 64)(7 62 34 65)(8 105 21 84)(9 99 15 78)(10 100 16 79)(11 101 17 80)(12 102 18 81)(13 103 19 82)(14 104 20 83)(22 89 111 92)(23 90 112 93)(24 91 106 94)(25 85 107 95)(26 86 108 96)(27 87 109 97)(28 88 110 98)(36 75 47 54)(37 76 48 55)(38 77 49 56)(39 71 43 50)(40 72 44 51)(41 73 45 52)(42 74 46 53)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 112)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 104)(93 105)(94 99)(95 100)(96 101)(97 102)(98 103)
(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)

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

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

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

175 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4H 4I ··· 4O 7A ··· 7F 14A ··· 14R 14S ··· 14AP 14AQ ··· 14BB 28A ··· 28AV 28AW ··· 28CL order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 2 2 4 4 2 ··· 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

175 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 C14 C14 C4○D4 C7×C4○D4 2+ 1+4 C7×2+ 1+4 kernel C7×C22.45C24 C14×C22⋊C4 C7×C42⋊C2 D4×C28 C7×C22≀C2 C7×C22⋊Q8 C7×C22.D4 C7×C4.4D4 C7×C42⋊2C2 C22.45C24 C2×C22⋊C4 C42⋊C2 C4×D4 C22≀C2 C22⋊Q8 C22.D4 C4.4D4 C42⋊2C2 C2×C14 C22 C14 C2 # reps 1 2 2 2 1 2 3 1 2 6 12 12 12 6 12 18 6 12 8 48 1 6

Matrix representation of C7×C22.45C24 in GL4(𝔽29) generated by

 23 0 0 0 0 23 0 0 0 0 25 0 0 0 0 25
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 12 0 0 0 0 17 0 0 0 0 28 27 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 12 0 0 0 0 12
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 28 28
,
 1 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,25,0,0,0,0,25],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[12,0,0,0,0,17,0,0,0,0,28,0,0,0,27,1],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[28,0,0,0,0,28,0,0,0,0,1,28,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1] >;

C7×C22.45C24 in GAP, Magma, Sage, TeX

C_7\times C_2^2._{45}C_2^4
% in TeX

G:=Group("C7xC2^2.45C2^4");
// GroupNames label

G:=SmallGroup(448,1334);
// by ID

G=gap.SmallGroup(448,1334);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,1576,4790,1690]);
// Polycyclic

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

׿
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