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G = C7×C22.29C24  order 448 = 26·7

Direct product of C7 and C22.29C24

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

Aliases: C7×C22.29C24, C14.1522+ (1+4), (C2×C28)⋊26D4, C4⋊D4⋊6C14, C4⋊1D4⋊5C14, C42⋊6(C2×C14), C4.16(D4×C14), C22≀C2⋊3C14, (C4×C28)⋊40C22, C4.4D4⋊6C14, C28.323(C2×D4), (C22×D4)⋊7C14, (D4×C14)⋊36C22, C24.19(C2×C14), (Q8×C14)⋊51C22, C22.21(D4×C14), C42⋊C2⋊10C14, (C2×C28).664C23, (C2×C14).355C24, (C22×C28)⋊48C22, C14.190(C22×D4), C23.9(C22×C14), C2.4(C7×2+ (1+4)), (C23×C14).16C22, C22.29(C23×C14), (C22×C14).91C23, (C2×C4)⋊4(C7×D4), (D4×C2×C14)⋊22C2, C4⋊C4⋊14(C2×C14), C2.14(D4×C2×C14), (C2×C4○D4)⋊4C14, (C2×D4)⋊4(C2×C14), (C14×C4○D4)⋊20C2, (C7×C4⋊1D4)⋊16C2, (C7×C4⋊D4)⋊33C2, C22⋊C4⋊4(C2×C14), (C7×C4⋊C4)⋊70C22, (C22×C4)⋊8(C2×C14), (C2×Q8)⋊11(C2×C14), (C7×C22≀C2)⋊13C2, (C2×C14).417(C2×D4), (C7×C4.4D4)⋊26C2, (C7×C42⋊C2)⋊31C2, (C7×C22⋊C4)⋊39C22, (C2×C4).22(C22×C14), SmallGroup(448,1318)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.29C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C4⋊1D4 — C7×C22.29C24
Lower central
C1 — C22 — C7×C22.29C24
Upper central
C1 — C2×C14 — C7×C22.29C24

Subgroups: 610 in 334 conjugacy classes, 162 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C7, C2×C4 [×2], C2×C4 [×10], C2×C4 [×4], D4 [×22], Q8 [×2], C23, C23 [×6], C23 [×8], C14, C14 [×2], C14 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×14], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24 [×2], C28 [×4], C28 [×6], C2×C14, C2×C14 [×2], C2×C14 [×28], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, C2×C28 [×2], C2×C28 [×10], C2×C28 [×4], C7×D4 [×22], C7×Q8 [×2], C22×C14, C22×C14 [×6], C22×C14 [×8], C22.29C24, C4×C28 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×2], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×14], D4×C14 [×4], Q8×C14, C7×C4○D4 [×4], C23×C14 [×2], C7×C42⋊C2, C7×C22≀C2 [×4], C7×C4⋊D4 [×4], C7×C4.4D4 [×2], C7×C4⋊1D4 [×2], D4×C2×C14, C14×C4○D4, C7×C22.29C24

Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C24, C2×C14 [×35], C22×D4, 2+ (1+4) [×2], C7×D4 [×4], C22×C14 [×15], C22.29C24, D4×C14 [×6], C23×C14, D4×C2×C14, C7×2+ (1+4) [×2], C7×C22.29C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, 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 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)(8 105 23 90)(9 99 24 91)(10 100 25 85)(11 101 26 86)(12 102 27 87)(13 103 28 88)(14 104 22 89)(15 78 106 94)(16 79 107 95)(17 80 108 96)(18 81 109 97)(19 82 110 98)(20 83 111 92)(21 84 112 93)(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)

(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)(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 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 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(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)(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 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), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,105,23,90)(9,99,24,91)(10,100,25,85)(11,101,26,86)(12,102,27,87)(13,103,28,88)(14,104,22,89)(15,78,106,94)(16,79,107,95)(17,80,108,96)(18,81,109,97)(19,82,110,98)(20,83,111,92)(21,84,112,93)(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), (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)(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)(78,99)(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), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(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)(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)(78,91)(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), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,105,23,90)(9,99,24,91)(10,100,25,85)(11,101,26,86)(12,102,27,87)(13,103,28,88)(14,104,22,89)(15,78,106,94)(16,79,107,95)(17,80,108,96)(18,81,109,97)(19,82,110,98)(20,83,111,92)(21,84,112,93)(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), (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)(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)(78,99)(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), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(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)(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)(78,91)(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)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,105,23,90),(9,99,24,91),(10,100,25,85),(11,101,26,86),(12,102,27,87),(13,103,28,88),(14,104,22,89),(15,78,106,94),(16,79,107,95),(17,80,108,96),(18,81,109,97),(19,82,110,98),(20,83,111,92),(21,84,112,93),(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)], [(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),(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),(78,99),(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)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(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),(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),(78,91),(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 ⊆ GL6(𝔽29)

100000
010000
007000
000700
000070
000007
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
0028000
0002800
0000280
0000028
,
0280000
2800000
000010
0005027
001000
00012024
,
2800000
0280000
001100
00272800
00024282
002424281
,
100000
0280000
001000
00272800
0000280
002424281
,
100000
010000
0028000
0002800
000010
0002401

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5,0,12,0,0,1,0,0,0,0,0,0,27,0,24],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,27,0,24,0,0,1,28,24,24,0,0,0,0,28,28,0,0,0,0,2,1],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,27,0,24,0,0,0,28,0,24,0,0,0,0,28,28,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,24,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

154 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4J7A···7F14A···14R14S···14AD14AE···14BN28A···28X28Y···28BH
order1222222···244444···47···714···1414···1414···1428···2828···28
size1111224···422224···41···11···12···24···42···24···4

154 irreducible representations

dim11111111111111112244
type++++++++++
imageC1C2C2C2C2C2C2C2C7C14C14C14C14C14C14C14D4C7×D42+ (1+4)C7×2+ (1+4)
kernelC7×C22.29C24C7×C42⋊C2C7×C22≀C2C7×C4⋊D4C7×C4.4D4C7×C4⋊1D4D4×C2×C14C14×C4○D4C22.29C24C42⋊C2C22≀C2C4⋊D4C4.4D4C4⋊1D4C22×D4C2×C4○D4C2×C28C2×C4C14C2
# reps11442211662424121266424212

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{29}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.29C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1318);
 
// by ID
 
G=gap.SmallGroup(448,1318);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,792,4790,1227,3363]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=f^2=g^2=1,e^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^-1=g*d*g=b*d=d*b,f*e*f=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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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