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## G = Q8×C3×C6order 144 = 24·32

### Direct product of C3×C6 and Q8

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

Aliases: Q8×C3×C6, C4.4C62, C22.4C62, C62.38C22, (C2×C12).15C6, (C6×C12).15C2, C12.26(C2×C6), C2.2(C2×C62), C6.15(C22×C6), (C3×C6).40C23, (C3×C12).53C22, (C2×C4).3(C3×C6), (C2×C6).21(C2×C6), SmallGroup(144,180)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C3×C6
 Chief series C1 — C2 — C6 — C3×C6 — C3×C12 — Q8×C32 — Q8×C3×C6
 Lower central C1 — C2 — Q8×C3×C6
 Upper central C1 — C62 — Q8×C3×C6

Generators and relations for Q8×C3×C6
G = < a,b,c,d | a3=b6=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 114, all normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, Q8, C32, C12, C2×C6, C2×Q8, C3×C6, C3×C6, C2×C12, C3×Q8, C3×C12, C62, C6×Q8, C6×C12, Q8×C32, Q8×C3×C6
Quotients: C1, C2, C3, C22, C6, Q8, C23, C32, C2×C6, C2×Q8, C3×C6, C3×Q8, C22×C6, C62, C6×Q8, Q8×C32, C2×C62, Q8×C3×C6

Smallest permutation representation of Q8×C3×C6
Regular action on 144 points
Generators in S144
(1 40 35)(2 41 36)(3 42 31)(4 37 32)(5 38 33)(6 39 34)(7 17 28)(8 18 29)(9 13 30)(10 14 25)(11 15 26)(12 16 27)(19 143 122)(20 144 123)(21 139 124)(22 140 125)(23 141 126)(24 142 121)(43 56 64)(44 57 65)(45 58 66)(46 59 61)(47 60 62)(48 55 63)(49 75 70)(50 76 71)(51 77 72)(52 78 67)(53 73 68)(54 74 69)(79 92 100)(80 93 101)(81 94 102)(82 95 97)(83 96 98)(84 91 99)(85 111 106)(86 112 107)(87 113 108)(88 114 103)(89 109 104)(90 110 105)(115 128 136)(116 129 137)(117 130 138)(118 131 133)(119 132 134)(120 127 135)
(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 53 10 55)(2 54 11 56)(3 49 12 57)(4 50 7 58)(5 51 8 59)(6 52 9 60)(13 62 39 78)(14 63 40 73)(15 64 41 74)(16 65 42 75)(17 66 37 76)(18 61 38 77)(19 109 138 99)(20 110 133 100)(21 111 134 101)(22 112 135 102)(23 113 136 97)(24 114 137 98)(25 48 35 68)(26 43 36 69)(27 44 31 70)(28 45 32 71)(29 46 33 72)(30 47 34 67)(79 144 105 118)(80 139 106 119)(81 140 107 120)(82 141 108 115)(83 142 103 116)(84 143 104 117)(85 132 93 124)(86 127 94 125)(87 128 95 126)(88 129 96 121)(89 130 91 122)(90 131 92 123)
(1 89 10 91)(2 90 11 92)(3 85 12 93)(4 86 7 94)(5 87 8 95)(6 88 9 96)(13 98 39 114)(14 99 40 109)(15 100 41 110)(16 101 42 111)(17 102 37 112)(18 97 38 113)(19 63 138 73)(20 64 133 74)(21 65 134 75)(22 66 135 76)(23 61 136 77)(24 62 137 78)(25 84 35 104)(26 79 36 105)(27 80 31 106)(28 81 32 107)(29 82 33 108)(30 83 34 103)(43 118 69 144)(44 119 70 139)(45 120 71 140)(46 115 72 141)(47 116 67 142)(48 117 68 143)(49 124 57 132)(50 125 58 127)(51 126 59 128)(52 121 60 129)(53 122 55 130)(54 123 56 131)

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

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

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

Q8×C3×C6 is a maximal subgroup of   C62.117D4  (C6×C12).C4  C62.134D4  C62.259C23  C62.261C23  C62.262C23  C3272- 1+4

90 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A ··· 4F 6A ··· 6X 12A ··· 12AV order 1 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C3 C6 C6 Q8 C3×Q8 kernel Q8×C3×C6 C6×C12 Q8×C32 C6×Q8 C2×C12 C3×Q8 C3×C6 C6 # reps 1 3 4 8 24 32 2 16

Matrix representation of Q8×C3×C6 in GL4(𝔽13) generated by

 9 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 10 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 0 12 0 0 1 0
,
 12 0 0 0 0 12 0 0 0 0 10 9 0 0 9 3
G:=sub<GL(4,GF(13))| [9,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[10,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,10,9,0,0,9,3] >;

Q8×C3×C6 in GAP, Magma, Sage, TeX

Q_8\times C_3\times C_6
% in TeX

G:=Group("Q8xC3xC6");
// GroupNames label

G:=SmallGroup(144,180);
// by ID

G=gap.SmallGroup(144,180);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-2,432,889,439]);
// Polycyclic

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

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