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G = C7×C22.54C24  order 448 = 26·7

Direct product of C7 and C22.54C24

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

Aliases: C7×C22.54C24, C14.1692+ (1+4), C4⋊1D4⋊9C14, C22≀C2⋊7C14, C4⋊D4⋊17C14, C42⋊11(C2×C14), (C4×C28)⋊45C22, C42⋊2C2⋊8C14, (D4×C14)⋊39C22, C24.24(C2×C14), (C2×C14).380C24, (C2×C28).681C23, (C22×C28)⋊52C22, C22.D4⋊13C14, C22.54(C23×C14), (C23×C14).21C22, C23.23(C22×C14), C2.21(C7×2+ (1+4)), (C22×C14).106C23, C4⋊C4⋊6(C2×C14), (C2×D4)⋊6(C2×C14), (C7×C4⋊D4)⋊44C2, (C7×C4⋊1D4)⋊20C2, (C7×C4⋊C4)⋊39C22, C22⋊C4⋊7(C2×C14), (C7×C22≀C2)⋊17C2, (C22×C4)⋊12(C2×C14), (C7×C42⋊2C2)⋊19C2, (C7×C22⋊C4)⋊42C22, (C2×C4).40(C22×C14), (C7×C22.D4)⋊32C2, SmallGroup(448,1343)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.54C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C4⋊1D4 — C7×C22.54C24
Lower central
C1 — C22 — C7×C22.54C24
Upper central
C1 — C2×C14 — C7×C22.54C24

Subgroups: 474 in 252 conjugacy classes, 142 normal (14 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C7, C2×C4 [×9], C2×C4 [×3], D4 [×12], C23, C23 [×5], C23 [×3], C14 [×3], C14 [×6], C42, C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×12], C24, C28 [×9], C2×C14, C2×C14 [×22], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2 [×2], C4⋊1D4, C2×C28 [×9], C2×C28 [×3], C7×D4 [×12], C22×C14, C22×C14 [×5], C22×C14 [×3], C22.54C24, C4×C28, C7×C22⋊C4 [×12], C7×C4⋊C4 [×6], C22×C28 [×3], D4×C14 [×12], C23×C14, C7×C22≀C2 [×3], C7×C4⋊D4 [×6], C7×C22.D4 [×3], C7×C42⋊2C2 [×2], C7×C4⋊1D4, C7×C22.54C24

Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C24, C2×C14 [×35], 2+ (1+4) [×3], C22×C14 [×15], C22.54C24, C23×C14, C7×2+ (1+4) [×3], C7×C22.54C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef=bce, fg=gf >

Smallest permutation representation
►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)

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

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

Matrix representation ►G ⊆ GL8(𝔽29)

230000000
023000000
002300000
000230000
000024000
000002400
000000240
000000024
,
280000000
028000000
002800000
000280000
00001000
00000100
00000010
00000001
,
280000000
028000000
002800000
000280000
000028000
000002800
000000280
000000028
,
102700000
002810000
002800000
012800000
000021010
000021001
000024080
000023180
,
127000000
028000000
1280280000
1282800000
00000100
00001000
000021801
000021810
,
10000000
01000000
102800000
100280000
000028000
00000100
000013010
000000028
,
10000000
128000000
00100000
100280000
00001000
00000100
0000160280
0000160028

G:=sub<GL(8,GF(29))| [23,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,24],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,27,28,28,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,21,21,24,23,0,0,0,0,0,0,0,1,0,0,0,0,1,0,8,8,0,0,0,0,0,1,0,0],[1,0,1,1,0,0,0,0,27,28,28,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,21,21,0,0,0,0,1,0,8,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28],[1,1,0,1,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28] >;
 

133 conjugacy classes

class 1 2A2B2C2D···2I4A···4I7A···7F14A···14R14S···14BB28A···28BB
order12222···24···47···714···1414···1428···28
size11114···44···41···11···14···44···4

133 irreducible representations

dim11111111111144
type+++++++
imageC1C2C2C2C2C2C7C14C14C14C14C142+ (1+4)C7×2+ (1+4)
kernelC7×C22.54C24C7×C22≀C2C7×C4⋊D4C7×C22.D4C7×C42⋊2C2C7×C4⋊1D4C22.54C24C22≀C2C4⋊D4C22.D4C42⋊2C2C4⋊1D4C14C2
# reps1363216183618126318

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{54}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.54C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1343);
 
// by ID
 
G=gap.SmallGroup(448,1343);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,3579,9635,1690]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=f^2=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,e*d*e=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,g*d*g=b*c*d,f*e*f=b*c*e,f*g=g*f>;
 
// generators/relations
 

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