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G = C7×C22.11C24  order 448 = 26·7

Direct product of C7 and C22.11C24

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

Aliases: C7×C22.11C24, C14.1492+ (1+4), (C4×D4)⋊5C14, D4⋊7(C2×C28), (C2×D4)⋊11C28, (D4×C14)⋊23C4, (D4×C28)⋊34C2, C42⋊4(C2×C14), C23⋊4(C2×C28), (C4×C28)⋊38C22, C2.7(C23×C28), C42⋊C2⋊6C14, C24.14(C2×C14), C4.19(C22×C28), C14.59(C23×C4), (C22×C28)⋊5C22, (C22×D4).9C14, (C2×C28).709C23, C28.164(C22×C4), (C2×C14).338C24, C22.2(C22×C28), C2.1(C7×2+ (1+4)), (D4×C14).332C22, C23.30(C22×C14), (C23×C14).11C22, C22.11(C23×C14), (C22×C14).254C23, (C2×C4)⋊4(C2×C28), C4⋊C4⋊20(C2×C14), (C2×C28)⋊25(C2×C4), (C7×D4)⋊27(C2×C4), (D4×C2×C14).22C2, (C2×C22⋊C4)⋊5C14, (C7×C4⋊C4)⋊77C22, (C22×C4)⋊3(C2×C14), (C22×C14)⋊5(C2×C4), C22⋊C4⋊18(C2×C14), (C14×C22⋊C4)⋊10C2, (C2×D4).78(C2×C14), (C7×C42⋊C2)⋊27C2, (C7×C22⋊C4)⋊72C22, (C2×C14).33(C22×C4), (C2×C4).56(C22×C14), SmallGroup(448,1301)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C7×C22.11C24
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C22⋊C4 — D4×C28 — C7×C22.11C24
Lower central
C1 — C2 — C7×C22.11C24
Upper central
C1 — C2×C14 — C7×C22.11C24

Subgroups: 514 in 338 conjugacy classes, 242 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C7, C2×C4 [×14], C2×C4 [×8], D4 [×16], C23, C23 [×12], C23 [×4], C14, C14 [×2], C14 [×10], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], C28 [×4], C28 [×8], C2×C14, C2×C14 [×10], C2×C14 [×18], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C2×C28 [×14], C2×C28 [×8], C7×D4 [×16], C22×C14, C22×C14 [×12], C22×C14 [×4], C22.11C24, C4×C28 [×4], C7×C22⋊C4 [×12], C7×C4⋊C4 [×4], C22×C28, C22×C28 [×8], D4×C14 [×12], C23×C14 [×2], C14×C22⋊C4 [×4], C7×C42⋊C2 [×2], D4×C28 [×8], D4×C2×C14, C7×C22.11C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C7, C2×C4 [×28], C23 [×15], C14 [×15], C22×C4 [×14], C24, C28 [×8], C2×C14 [×35], C23×C4, 2+ (1+4) [×2], C2×C28 [×28], C22×C14 [×15], C22.11C24, C22×C28 [×14], C23×C14, C23×C28, C7×2+ (1+4) [×2], C7×C22.11C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, 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 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 106)(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)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)

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

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

(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)

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

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

Matrix representation ►G ⊆ GL5(𝔽29)

10000
016000
001600
000160
000016
,
10000
028000
002800
000280
000028
,
280000
01000
00100
00010
00001
,
120000
017020
000281
010120
011120
,
280000
01000
0282800
0120280
00001
,
280000
01200
002800
001201
001210
,
10000
01000
00100
0120280
0120028

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[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],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,17,0,1,1,0,0,0,0,1,0,2,28,12,12,0,0,1,0,0],[28,0,0,0,0,0,1,28,12,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,2,28,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,12,12,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28] >;
 

238 conjugacy classes

class 1 2A2B2C2D···2M4A···4T7A···7F14A···14R14S···14BZ28A···28DP
order12222···24···47···714···1414···1428···28
size11112···22···21···11···12···22···2

238 irreducible representations

dim11111111111144
type++++++
imageC1C2C2C2C2C4C7C14C14C14C14C282+ (1+4)C7×2+ (1+4)
kernelC7×C22.11C24C14×C22⋊C4C7×C42⋊C2D4×C28D4×C2×C14D4×C14C22.11C24C2×C22⋊C4C42⋊C2C4×D4C22×D4C2×D4C14C2
# reps14281166241248696212

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{11}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.11C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1301);
 
// by ID
 
G=gap.SmallGroup(448,1301);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1227,3363]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=e^2=f^2=g^2=1,d^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=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*f=f*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀