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G = C626Q8order 288 = 25·32

4th semidirect product of C62 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C626Q8, C62.222C23, (C2×C6)⋊5Dic6, (C2×C12).29D6, C6.105(S3×D4), C3⋊Dic3.63D4, C6.41(C2×Dic6), (C22×C6).84D6, C12⋊Dic35C2, C625C4.5C2, (C6×C12).10C22, C6.Dic64C2, C3217(C22⋊Q8), C6.92(D42S3), (C2×C62).61C22, C222(C324Q8), C35(Dic3.D4), C2.6(C12.D6), C2.6(D4×C3⋊S3), (C3×C6).55(C2×Q8), (C3×C6).228(C2×D4), (C3×C22⋊C4).5S3, C23.17(C2×C3⋊S3), (C2×C324Q8)⋊4C2, C22⋊C4.1(C3⋊S3), C2.6(C2×C324Q8), (C3×C6).142(C4○D4), (C2×C6).239(C22×S3), (C32×C22⋊C4).1C2, C22.39(C22×C3⋊S3), (C22×C3⋊Dic3).10C2, (C2×C3⋊Dic3).78C22, (C2×C4).5(C2×C3⋊S3), SmallGroup(288,735)

Series: Derived Chief Lower central Upper central

C1C62 — C626Q8
C1C3C32C3×C6C62C2×C3⋊Dic3C22×C3⋊Dic3 — C626Q8
C32C62 — C626Q8
C1C22C22⋊C4

Generators and relations for C626Q8
 G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 732 in 222 conjugacy classes, 79 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×7], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C32, Dic3 [×20], C12 [×8], C2×C6 [×12], C2×C6 [×8], C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×8], C2×Dic3 [×24], C2×C12 [×8], C22×C6 [×4], C22⋊Q8, C3⋊Dic3 [×2], C3⋊Dic3 [×3], C3×C12 [×2], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12 [×2], C2×C62, Dic3.D4 [×4], C6.Dic6 [×2], C12⋊Dic3, C625C4, C32×C22⋊C4, C2×C324Q8, C22×C3⋊Dic3, C626Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], Q8 [×2], C23, D6 [×12], C2×D4, C2×Q8, C4○D4, C3⋊S3, Dic6 [×8], C22×S3 [×4], C22⋊Q8, C2×C3⋊S3 [×3], C2×Dic6 [×4], S3×D4 [×4], D42S3 [×4], C324Q8 [×2], C22×C3⋊S3, Dic3.D4 [×4], C2×C324Q8, D4×C3⋊S3, C12.D6, C626Q8

Smallest permutation representation of C626Q8
On 144 points
Generators in S144
(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)(127 128 129 130 131 132)(133 134 135 136 137 138)(139 140 141 142 143 144)
(1 7 38 20 24 4)(2 8 39 21 22 5)(3 9 37 19 23 6)(10 32 35 28 14 26)(11 33 36 29 15 27)(12 31 34 30 13 25)(16 61 59 57 66 68)(17 62 60 55 64 69)(18 63 58 56 65 67)(40 45 70 49 54 48)(41 43 71 50 52 46)(42 44 72 51 53 47)(73 80 143 76 83 140)(74 81 144 77 84 141)(75 82 139 78 79 142)(85 94 100 88 91 97)(86 95 101 89 92 98)(87 96 102 90 93 99)(103 118 111 106 115 114)(104 119 112 107 116 109)(105 120 113 108 117 110)(121 136 129 124 133 132)(122 137 130 125 134 127)(123 138 131 126 135 128)
(1 87 13 83)(2 85 14 81)(3 89 15 79)(4 99 30 76)(5 97 28 74)(6 101 29 78)(7 96 25 140)(8 94 26 144)(9 92 27 142)(10 77 39 100)(11 75 37 98)(12 73 38 102)(16 111 41 129)(17 109 42 127)(18 113 40 131)(19 86 33 82)(20 90 31 80)(21 88 32 84)(22 91 35 141)(23 95 36 139)(24 93 34 143)(43 124 61 106)(44 122 62 104)(45 126 63 108)(46 136 68 118)(47 134 69 116)(48 138 67 120)(49 128 56 110)(50 132 57 114)(51 130 55 112)(52 121 66 103)(53 125 64 107)(54 123 65 105)(58 117 70 135)(59 115 71 133)(60 119 72 137)
(1 46 13 68)(2 48 14 67)(3 47 15 69)(4 41 30 16)(5 40 28 18)(6 42 29 17)(7 52 25 66)(8 54 26 65)(9 53 27 64)(10 56 39 49)(11 55 37 51)(12 57 38 50)(19 72 33 60)(20 71 31 59)(21 70 32 58)(22 45 35 63)(23 44 36 62)(24 43 34 61)(73 132 102 114)(74 131 97 113)(75 130 98 112)(76 129 99 111)(77 128 100 110)(78 127 101 109)(79 134 89 116)(80 133 90 115)(81 138 85 120)(82 137 86 119)(83 136 87 118)(84 135 88 117)(91 108 141 126)(92 107 142 125)(93 106 143 124)(94 105 144 123)(95 104 139 122)(96 103 140 121)

G:=sub<Sym(144)| (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)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144), (1,7,38,20,24,4)(2,8,39,21,22,5)(3,9,37,19,23,6)(10,32,35,28,14,26)(11,33,36,29,15,27)(12,31,34,30,13,25)(16,61,59,57,66,68)(17,62,60,55,64,69)(18,63,58,56,65,67)(40,45,70,49,54,48)(41,43,71,50,52,46)(42,44,72,51,53,47)(73,80,143,76,83,140)(74,81,144,77,84,141)(75,82,139,78,79,142)(85,94,100,88,91,97)(86,95,101,89,92,98)(87,96,102,90,93,99)(103,118,111,106,115,114)(104,119,112,107,116,109)(105,120,113,108,117,110)(121,136,129,124,133,132)(122,137,130,125,134,127)(123,138,131,126,135,128), (1,87,13,83)(2,85,14,81)(3,89,15,79)(4,99,30,76)(5,97,28,74)(6,101,29,78)(7,96,25,140)(8,94,26,144)(9,92,27,142)(10,77,39,100)(11,75,37,98)(12,73,38,102)(16,111,41,129)(17,109,42,127)(18,113,40,131)(19,86,33,82)(20,90,31,80)(21,88,32,84)(22,91,35,141)(23,95,36,139)(24,93,34,143)(43,124,61,106)(44,122,62,104)(45,126,63,108)(46,136,68,118)(47,134,69,116)(48,138,67,120)(49,128,56,110)(50,132,57,114)(51,130,55,112)(52,121,66,103)(53,125,64,107)(54,123,65,105)(58,117,70,135)(59,115,71,133)(60,119,72,137), (1,46,13,68)(2,48,14,67)(3,47,15,69)(4,41,30,16)(5,40,28,18)(6,42,29,17)(7,52,25,66)(8,54,26,65)(9,53,27,64)(10,56,39,49)(11,55,37,51)(12,57,38,50)(19,72,33,60)(20,71,31,59)(21,70,32,58)(22,45,35,63)(23,44,36,62)(24,43,34,61)(73,132,102,114)(74,131,97,113)(75,130,98,112)(76,129,99,111)(77,128,100,110)(78,127,101,109)(79,134,89,116)(80,133,90,115)(81,138,85,120)(82,137,86,119)(83,136,87,118)(84,135,88,117)(91,108,141,126)(92,107,142,125)(93,106,143,124)(94,105,144,123)(95,104,139,122)(96,103,140,121)>;

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)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144), (1,7,38,20,24,4)(2,8,39,21,22,5)(3,9,37,19,23,6)(10,32,35,28,14,26)(11,33,36,29,15,27)(12,31,34,30,13,25)(16,61,59,57,66,68)(17,62,60,55,64,69)(18,63,58,56,65,67)(40,45,70,49,54,48)(41,43,71,50,52,46)(42,44,72,51,53,47)(73,80,143,76,83,140)(74,81,144,77,84,141)(75,82,139,78,79,142)(85,94,100,88,91,97)(86,95,101,89,92,98)(87,96,102,90,93,99)(103,118,111,106,115,114)(104,119,112,107,116,109)(105,120,113,108,117,110)(121,136,129,124,133,132)(122,137,130,125,134,127)(123,138,131,126,135,128), (1,87,13,83)(2,85,14,81)(3,89,15,79)(4,99,30,76)(5,97,28,74)(6,101,29,78)(7,96,25,140)(8,94,26,144)(9,92,27,142)(10,77,39,100)(11,75,37,98)(12,73,38,102)(16,111,41,129)(17,109,42,127)(18,113,40,131)(19,86,33,82)(20,90,31,80)(21,88,32,84)(22,91,35,141)(23,95,36,139)(24,93,34,143)(43,124,61,106)(44,122,62,104)(45,126,63,108)(46,136,68,118)(47,134,69,116)(48,138,67,120)(49,128,56,110)(50,132,57,114)(51,130,55,112)(52,121,66,103)(53,125,64,107)(54,123,65,105)(58,117,70,135)(59,115,71,133)(60,119,72,137), (1,46,13,68)(2,48,14,67)(3,47,15,69)(4,41,30,16)(5,40,28,18)(6,42,29,17)(7,52,25,66)(8,54,26,65)(9,53,27,64)(10,56,39,49)(11,55,37,51)(12,57,38,50)(19,72,33,60)(20,71,31,59)(21,70,32,58)(22,45,35,63)(23,44,36,62)(24,43,34,61)(73,132,102,114)(74,131,97,113)(75,130,98,112)(76,129,99,111)(77,128,100,110)(78,127,101,109)(79,134,89,116)(80,133,90,115)(81,138,85,120)(82,137,86,119)(83,136,87,118)(84,135,88,117)(91,108,141,126)(92,107,142,125)(93,106,143,124)(94,105,144,123)(95,104,139,122)(96,103,140,121) );

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),(127,128,129,130,131,132),(133,134,135,136,137,138),(139,140,141,142,143,144)], [(1,7,38,20,24,4),(2,8,39,21,22,5),(3,9,37,19,23,6),(10,32,35,28,14,26),(11,33,36,29,15,27),(12,31,34,30,13,25),(16,61,59,57,66,68),(17,62,60,55,64,69),(18,63,58,56,65,67),(40,45,70,49,54,48),(41,43,71,50,52,46),(42,44,72,51,53,47),(73,80,143,76,83,140),(74,81,144,77,84,141),(75,82,139,78,79,142),(85,94,100,88,91,97),(86,95,101,89,92,98),(87,96,102,90,93,99),(103,118,111,106,115,114),(104,119,112,107,116,109),(105,120,113,108,117,110),(121,136,129,124,133,132),(122,137,130,125,134,127),(123,138,131,126,135,128)], [(1,87,13,83),(2,85,14,81),(3,89,15,79),(4,99,30,76),(5,97,28,74),(6,101,29,78),(7,96,25,140),(8,94,26,144),(9,92,27,142),(10,77,39,100),(11,75,37,98),(12,73,38,102),(16,111,41,129),(17,109,42,127),(18,113,40,131),(19,86,33,82),(20,90,31,80),(21,88,32,84),(22,91,35,141),(23,95,36,139),(24,93,34,143),(43,124,61,106),(44,122,62,104),(45,126,63,108),(46,136,68,118),(47,134,69,116),(48,138,67,120),(49,128,56,110),(50,132,57,114),(51,130,55,112),(52,121,66,103),(53,125,64,107),(54,123,65,105),(58,117,70,135),(59,115,71,133),(60,119,72,137)], [(1,46,13,68),(2,48,14,67),(3,47,15,69),(4,41,30,16),(5,40,28,18),(6,42,29,17),(7,52,25,66),(8,54,26,65),(9,53,27,64),(10,56,39,49),(11,55,37,51),(12,57,38,50),(19,72,33,60),(20,71,31,59),(21,70,32,58),(22,45,35,63),(23,44,36,62),(24,43,34,61),(73,132,102,114),(74,131,97,113),(75,130,98,112),(76,129,99,111),(77,128,100,110),(78,127,101,109),(79,134,89,116),(80,133,90,115),(81,138,85,120),(82,137,86,119),(83,136,87,118),(84,135,88,117),(91,108,141,126),(92,107,142,125),(93,106,143,124),(94,105,144,123),(95,104,139,122),(96,103,140,121)])

54 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D4E4F4G4H6A···6L6M···6T12A···12P
order1222223333444444446···66···612···12
size1111222222441818181836362···24···44···4

54 irreducible representations

dim1111111222222244
type+++++++++-++-+-
imageC1C2C2C2C2C2C2S3D4Q8D6D6C4○D4Dic6S3×D4D42S3
kernelC626Q8C6.Dic6C12⋊Dic3C625C4C32×C22⋊C4C2×C324Q8C22×C3⋊Dic3C3×C22⋊C4C3⋊Dic3C62C2×C12C22×C6C3×C6C2×C6C6C6
# reps12111114228421644

Matrix representation of C626Q8 in GL8(𝔽13)

10000000
012000000
00900000
00030000
0000121200
00001000
00000010
000000612
,
120000000
012000000
00900000
00030000
00001000
00000100
000000120
000000012
,
01000000
10000000
001200000
000120000
00001000
00000100
00000043
00000039
,
10000000
01000000
00010000
00100000
00001100
000001200
00000050
00000048

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,5,4,0,0,0,0,0,0,0,8] >;

C626Q8 in GAP, Magma, Sage, TeX

C_6^2\rtimes_6Q_8
% in TeX

G:=Group("C6^2:6Q8");
// GroupNames label

G:=SmallGroup(288,735);
// by ID

G=gap.SmallGroup(288,735);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,254,219,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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