direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C2.C25, C14.26C25, C28.95C24, 2+ (1+4)⋊5C14, 2- (1+4)⋊4C14, C2.6(C24×C14), (D4×C14)⋊69C22, C4.15(C23×C14), (C2×C14).10C24, (Q8×C14)⋊58C22, D4.9(C22×C14), (C7×D4).42C23, Q8.9(C22×C14), (C7×Q8).43C23, (C2×C28).691C23, (C22×C28)⋊54C22, C22.4(C23×C14), (C7×2- (1+4))⋊9C2, C23.27(C22×C14), (C7×2+ (1+4))⋊11C2, (C22×C14).110C23, C4○D4⋊8(C2×C14), (C2×C4○D4)⋊15C14, (C14×C4○D4)⋊31C2, (C2×D4)⋊18(C2×C14), (C2×Q8)⋊18(C2×C14), (C22×C4)⋊14(C2×C14), (C7×C4○D4)⋊28C22, (C2×C4).52(C22×C14), SmallGroup(448,1391)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 930 in 810 conjugacy classes, 750 normal (8 characteristic)
C1, C2, C2 [×15], C4, C4 [×15], C22 [×15], C22 [×15], C7, C2×C4 [×60], D4 [×60], Q8 [×20], C23 [×15], C14, C14 [×15], C22×C4 [×15], C2×D4 [×45], C2×Q8 [×15], C4○D4 [×80], C28, C28 [×15], C2×C14 [×15], C2×C14 [×15], C2×C4○D4 [×15], 2+ (1+4) [×10], 2- (1+4) [×6], C2×C28 [×60], C7×D4 [×60], C7×Q8 [×20], C22×C14 [×15], C2.C25, C22×C28 [×15], D4×C14 [×45], Q8×C14 [×15], C7×C4○D4 [×80], C14×C4○D4 [×15], C7×2+ (1+4) [×10], C7×2- (1+4) [×6], C7×C2.C25
Quotients:
C1, C2 [×31], C22 [×155], C7, C23 [×155], C14 [×31], C24 [×31], C2×C14 [×155], C25, C22×C14 [×155], C2.C25, C23×C14 [×31], C24×C14, C7×C2.C25
Generators and relations
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, dcd=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >
►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 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 111)(9 112)(10 106)(11 107)(12 108)(13 109)(14 110)(22 31)(23 32)(24 33)(25 34)(26 35)(27 29)(28 30)(36 45)(37 46)(38 47)(39 48)(40 49)(41 43)(42 44)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 101)(93 102)(94 103)(95 104)(96 105)(97 99)(98 100)
(36 45)(37 46)(38 47)(39 48)(40 49)(41 43)(42 44)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 64)(7 65)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(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)
(8 111)(9 112)(10 106)(11 107)(12 108)(13 109)(14 110)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 101)(93 102)(94 103)(95 104)(96 105)(97 99)(98 100)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 33 19 24)(2 34 20 25)(3 35 21 26)(4 29 15 27)(5 30 16 28)(6 31 17 22)(7 32 18 23)(8 99 111 97)(9 100 112 98)(10 101 106 92)(11 102 107 93)(12 103 108 94)(13 104 109 95)(14 105 110 96)(36 59 45 50)(37 60 46 51)(38 61 47 52)(39 62 48 53)(40 63 49 54)(41 57 43 55)(42 58 44 56)(64 87 73 78)(65 88 74 79)(66 89 75 80)(67 90 76 81)(68 91 77 82)(69 85 71 83)(70 86 72 84)
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), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,111)(9,112)(10,106)(11,107)(12,108)(13,109)(14,110)(22,31)(23,32)(24,33)(25,34)(26,35)(27,29)(28,30)(36,45)(37,46)(38,47)(39,48)(40,49)(41,43)(42,44)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (36,45)(37,46)(38,47)(39,48)(40,49)(41,43)(42,44)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86), (1,66)(2,67)(3,68)(4,69)(5,70)(6,64)(7,65)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(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), (8,111)(9,112)(10,106)(11,107)(12,108)(13,109)(14,110)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,33,19,24)(2,34,20,25)(3,35,21,26)(4,29,15,27)(5,30,16,28)(6,31,17,22)(7,32,18,23)(8,99,111,97)(9,100,112,98)(10,101,106,92)(11,102,107,93)(12,103,108,94)(13,104,109,95)(14,105,110,96)(36,59,45,50)(37,60,46,51)(38,61,47,52)(39,62,48,53)(40,63,49,54)(41,57,43,55)(42,58,44,56)(64,87,73,78)(65,88,74,79)(66,89,75,80)(67,90,76,81)(68,91,77,82)(69,85,71,83)(70,86,72,84)>;
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), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,111)(9,112)(10,106)(11,107)(12,108)(13,109)(14,110)(22,31)(23,32)(24,33)(25,34)(26,35)(27,29)(28,30)(36,45)(37,46)(38,47)(39,48)(40,49)(41,43)(42,44)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (36,45)(37,46)(38,47)(39,48)(40,49)(41,43)(42,44)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86), (1,66)(2,67)(3,68)(4,69)(5,70)(6,64)(7,65)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(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), (8,111)(9,112)(10,106)(11,107)(12,108)(13,109)(14,110)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,33,19,24)(2,34,20,25)(3,35,21,26)(4,29,15,27)(5,30,16,28)(6,31,17,22)(7,32,18,23)(8,99,111,97)(9,100,112,98)(10,101,106,92)(11,102,107,93)(12,103,108,94)(13,104,109,95)(14,105,110,96)(36,59,45,50)(37,60,46,51)(38,61,47,52)(39,62,48,53)(40,63,49,54)(41,57,43,55)(42,58,44,56)(64,87,73,78)(65,88,74,79)(66,89,75,80)(67,90,76,81)(68,91,77,82)(69,85,71,83)(70,86,72,84) );
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)], [(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,111),(9,112),(10,106),(11,107),(12,108),(13,109),(14,110),(22,31),(23,32),(24,33),(25,34),(26,35),(27,29),(28,30),(36,45),(37,46),(38,47),(39,48),(40,49),(41,43),(42,44),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,73),(65,74),(66,75),(67,76),(68,77),(69,71),(70,72),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,101),(93,102),(94,103),(95,104),(96,105),(97,99),(98,100)], [(36,45),(37,46),(38,47),(39,48),(40,49),(41,43),(42,44),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,73),(65,74),(66,75),(67,76),(68,77),(69,71),(70,72),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,64),(7,65),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(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)], [(8,111),(9,112),(10,106),(11,107),(12,108),(13,109),(14,110),(64,73),(65,74),(66,75),(67,76),(68,77),(69,71),(70,72),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,101),(93,102),(94,103),(95,104),(96,105),(97,99),(98,100)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(1,33,19,24),(2,34,20,25),(3,35,21,26),(4,29,15,27),(5,30,16,28),(6,31,17,22),(7,32,18,23),(8,99,111,97),(9,100,112,98),(10,101,106,92),(11,102,107,93),(12,103,108,94),(13,104,109,95),(14,105,110,96),(36,59,45,50),(37,60,46,51),(38,61,47,52),(39,62,48,53),(40,63,49,54),(41,57,43,55),(42,58,44,56),(64,87,73,78),(65,88,74,79),(66,89,75,80),(67,90,76,81),(68,91,77,82),(69,85,71,83),(70,86,72,84)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,20,0,0,0,0,20],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,28,0,0,0,0,1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;
238 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2P | 4A | 4B | 4C | ··· | 4Q | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14CR | 28A | ··· | 28L | 28M | ··· | 28CX |
| order | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
238 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C2.C25 | C7×C2.C25 |
| kernel | C7×C2.C25 | C14×C4○D4 | C7×2+ (1+4) | C7×2- (1+4) | C2.C25 | C2×C4○D4 | 2+ (1+4) | 2- (1+4) | C7 | C1 |
| # reps | 1 | 15 | 10 | 6 | 6 | 90 | 60 | 36 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_7\times C_2.C_2^5
% in TeX
G:=Group("C7xC2.C2^5"); // GroupNames label
G:=SmallGroup(448,1391);
// by ID
G=gap.SmallGroup(448,1391);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-7,-2,3165,2403,6499,522]);
// 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,d*c*d=f*c*f=b*c=c*b,e*d*e=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*e=e*c,c*g=g*c,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀