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## G = C7×C22.19C24order 448 = 26·7

### Direct product of C7 and C22.19C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C22.19C24
 Chief series C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C22≀C2 — C7×C22.19C24
 Lower central C1 — C22 — C7×C22.19C24
 Upper central C1 — C2×C28 — C7×C22.19C24

Generators and relations for C7×C22.19C24
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 498 in 330 conjugacy classes, 170 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22×C14, C22.19C24, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C23×C14, C7×C42⋊C2, D4×C28, C7×C22≀C2, C7×C4⋊D4, C7×C22⋊Q8, C7×C22.D4, C23×C28, C14×C4○D4, C7×C22.19C24
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C24, C2×C14, C22×D4, C2×C4○D4, C7×D4, C22×C14, C22.19C24, D4×C14, C7×C4○D4, C23×C14, D4×C2×C14, C14×C4○D4, C7×C22.19C24

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

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

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

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

196 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 7A ··· 7F 14A ··· 14R 14S ··· 14BB 14BC ··· 14BN 28A ··· 28X 28Y ··· 28BH 28BI ··· 28CR order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 C14 C14 D4 C4○D4 C7×D4 C7×C4○D4 kernel C7×C22.19C24 C7×C42⋊C2 D4×C28 C7×C22≀C2 C7×C4⋊D4 C7×C22⋊Q8 C7×C22.D4 C23×C28 C14×C4○D4 C22.19C24 C42⋊C2 C4×D4 C22≀C2 C4⋊D4 C22⋊Q8 C22.D4 C23×C4 C2×C4○D4 C2×C28 C2×C14 C2×C4 C22 # reps 1 1 4 2 2 2 2 1 1 6 6 24 12 12 12 12 6 6 4 8 24 48

Matrix representation of C7×C22.19C24 in GL5(𝔽29)

 7 0 0 0 0 0 1 0 0 0 0 0 1 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 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 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 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 1 0 0 0 0 0 1
,
 28 0 0 0 0 0 17 0 0 0 0 0 17 0 0 0 0 0 17 0 0 0 0 0 17

G:=sub<GL(5,GF(29))| [7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,17] >;

C7×C22.19C24 in GAP, Magma, Sage, TeX

C_7\times C_2^2._{19}C_2^4
% in TeX

G:=Group("C7xC2^2.19C2^4");
// GroupNames label

G:=SmallGroup(448,1308);
// by ID

G=gap.SmallGroup(448,1308);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,416]);
// Polycyclic

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

׿
×
𝔽