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G = C21.A4order 252 = 22·32·7

The non-split extension by C21 of A4 acting via A4/C22=C3

metabelian, soluble, monomial, A-group

Aliases: C21.A4, C3.(C7⋊A4), C7⋊(C3.A4), C22⋊(C7⋊C9), (C2×C14)⋊2C9, (C2×C42).2C3, (C2×C6).(C7⋊C3), SmallGroup(252,11)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C21.A4
C1C7C2×C14C2×C42 — C21.A4
C2×C14 — C21.A4
C1C3

Generators and relations for C21.A4
 G = < a,b,c,d | a21=b2=c2=1, d3=a7, ab=ba, ac=ca, dad-1=a4, dbd-1=bc=cb, dcd-1=b >

3C2
3C6
28C9
3C14
3C42
4C7⋊C9
7C3.A4

Smallest permutation representation of C21.A4
On 126 points
Generators in S126
(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)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)
(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)(85 125)(86 126)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)(101 120)(102 121)(103 122)(104 123)(105 124)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 22)(19 23)(20 24)(21 25)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)
(1 111 44 8 118 51 15 125 58)(2 106 48 9 113 55 16 120 62)(3 122 52 10 108 59 17 115 45)(4 117 56 11 124 63 18 110 49)(5 112 60 12 119 46 19 126 53)(6 107 43 13 114 50 20 121 57)(7 123 47 14 109 54 21 116 61)(22 91 81 29 98 67 36 105 74)(23 86 64 30 93 71 37 100 78)(24 102 68 31 88 75 38 95 82)(25 97 72 32 104 79 39 90 65)(26 92 76 33 99 83 40 85 69)(27 87 80 34 94 66 41 101 73)(28 103 84 35 89 70 42 96 77)

G:=sub<Sym(126)| (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(85,125)(86,126)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,22)(19,23)(20,24)(21,25)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74), (1,111,44,8,118,51,15,125,58)(2,106,48,9,113,55,16,120,62)(3,122,52,10,108,59,17,115,45)(4,117,56,11,124,63,18,110,49)(5,112,60,12,119,46,19,126,53)(6,107,43,13,114,50,20,121,57)(7,123,47,14,109,54,21,116,61)(22,91,81,29,98,67,36,105,74)(23,86,64,30,93,71,37,100,78)(24,102,68,31,88,75,38,95,82)(25,97,72,32,104,79,39,90,65)(26,92,76,33,99,83,40,85,69)(27,87,80,34,94,66,41,101,73)(28,103,84,35,89,70,42,96,77)>;

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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(85,125)(86,126)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,22)(19,23)(20,24)(21,25)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74), (1,111,44,8,118,51,15,125,58)(2,106,48,9,113,55,16,120,62)(3,122,52,10,108,59,17,115,45)(4,117,56,11,124,63,18,110,49)(5,112,60,12,119,46,19,126,53)(6,107,43,13,114,50,20,121,57)(7,123,47,14,109,54,21,116,61)(22,91,81,29,98,67,36,105,74)(23,86,64,30,93,71,37,100,78)(24,102,68,31,88,75,38,95,82)(25,97,72,32,104,79,39,90,65)(26,92,76,33,99,83,40,85,69)(27,87,80,34,94,66,41,101,73)(28,103,84,35,89,70,42,96,77) );

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),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74),(85,125),(86,126),(87,106),(88,107),(89,108),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114),(96,115),(97,116),(98,117),(99,118),(100,119),(101,120),(102,121),(103,122),(104,123),(105,124)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,22),(19,23),(20,24),(21,25),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74)], [(1,111,44,8,118,51,15,125,58),(2,106,48,9,113,55,16,120,62),(3,122,52,10,108,59,17,115,45),(4,117,56,11,124,63,18,110,49),(5,112,60,12,119,46,19,126,53),(6,107,43,13,114,50,20,121,57),(7,123,47,14,109,54,21,116,61),(22,91,81,29,98,67,36,105,74),(23,86,64,30,93,71,37,100,78),(24,102,68,31,88,75,38,95,82),(25,97,72,32,104,79,39,90,65),(26,92,76,33,99,83,40,85,69),(27,87,80,34,94,66,41,101,73),(28,103,84,35,89,70,42,96,77)]])

36 conjugacy classes

class 1  2 3A3B6A6B7A7B9A···9F14A···14F21A21B21C21D42A···42L
order123366779···914···142121212142···42
size1311333328···283···333333···3

36 irreducible representations

dim111333333
type++
imageC1C3C9A4C7⋊C3C3.A4C7⋊C9C7⋊A4C21.A4
kernelC21.A4C2×C42C2×C14C21C2×C6C7C22C3C1
# reps1261224612

Matrix representation of C21.A4 in GL6(𝔽127)

202079000
7959107000
10700000
0003200
0007980
00064064
,
100000
010000
001000
000100
0001231260
0001230126
,
100000
010000
001000
00012600
00001260
000401
,
8161107000
224420000
10722000
0001231250
0001041
0007140

G:=sub<GL(6,GF(127))| [20,79,107,0,0,0,20,59,0,0,0,0,79,107,0,0,0,0,0,0,0,32,79,64,0,0,0,0,8,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,123,123,0,0,0,0,126,0,0,0,0,0,0,126],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,126,0,4,0,0,0,0,126,0,0,0,0,0,0,1],[81,22,107,0,0,0,61,44,2,0,0,0,107,20,2,0,0,0,0,0,0,123,10,71,0,0,0,125,4,4,0,0,0,0,1,0] >;

C21.A4 in GAP, Magma, Sage, TeX

C_{21}.A_4
% in TeX

G:=Group("C21.A4");
// GroupNames label

G:=SmallGroup(252,11);
// by ID

G=gap.SmallGroup(252,11);
# by ID

G:=PCGroup([5,-3,-3,-2,2,-7,15,272,543,1804]);
// Polycyclic

G:=Group<a,b,c,d|a^21=b^2=c^2=1,d^3=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of C21.A4 in TeX

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