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G = C2×C23.D7order 224 = 25·7

Direct product of C2 and C23.D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C23.D7
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×C23.D7
 Lower central C7 — C14 — C2×C23.D7
 Upper central C1 — C23 — C24

Generators and relations for C2×C23.D7
G = < a,b,c,d,e,f | a2=b2=c2=d2=e7=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 350 in 132 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C24, Dic7, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×Dic7, C2×Dic7, C22×C14, C22×C14, C22×C14, C23.D7, C22×Dic7, C23×C14, C2×C23.D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C23.D7

Smallest permutation representation of C2×C23.D7
On 112 points
Generators in S112
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(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)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(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 75 15 61)(2 74 16 60)(3 73 17 59)(4 72 18 58)(5 71 19 57)(6 77 20 63)(7 76 21 62)(8 82 22 68)(9 81 23 67)(10 80 24 66)(11 79 25 65)(12 78 26 64)(13 84 27 70)(14 83 28 69)(29 103 43 89)(30 102 44 88)(31 101 45 87)(32 100 46 86)(33 99 47 85)(34 105 48 91)(35 104 49 90)(36 110 50 96)(37 109 51 95)(38 108 52 94)(39 107 53 93)(40 106 54 92)(41 112 55 98)(42 111 56 97)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 7A 7B 7C 14A ··· 14AS order 1 2 ··· 2 2 2 2 2 4 ··· 4 7 7 7 14 ··· 14 size 1 1 ··· 1 2 2 2 2 14 ··· 14 2 2 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + + - + image C1 C2 C2 C2 C4 D4 D7 Dic7 D14 C7⋊D4 kernel C2×C23.D7 C23.D7 C22×Dic7 C23×C14 C22×C14 C2×C14 C24 C23 C23 C22 # reps 1 4 2 1 8 4 3 12 9 24

Matrix representation of C2×C23.D7 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 28
,
 28 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 24 0 0 0 0 23
,
 17 0 0 0 0 1 0 0 0 0 0 23 0 0 5 0
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28],[28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,24,0,0,0,0,23],[17,0,0,0,0,1,0,0,0,0,0,5,0,0,23,0] >;

C2×C23.D7 in GAP, Magma, Sage, TeX

C_2\times C_2^3.D_7
% in TeX

G:=Group("C2xC2^3.D7");
// GroupNames label

G:=SmallGroup(224,147);
// by ID

G=gap.SmallGroup(224,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,362,6917]);
// Polycyclic

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

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