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## G = C2×He3⋊3Q8order 432 = 24·33

### Direct product of C2 and He3⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×He3⋊3Q8
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C32⋊C12 — C2×C32⋊C12 — C2×He3⋊3Q8
 Lower central C32 — C3×C6 — C2×He3⋊3Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×He33Q8
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf-1=b-1, cd=dc, ce=ec, fcf-1=c-1, de=ed, df=fd, fef-1=e-1 >

Subgroups: 537 in 149 conjugacy classes, 62 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C2×Dic6, C6×Q8, C2×He3, C2×He3, C3×Dic6, C6×Dic3, C324Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C32⋊C12, C4×He3, C22×He3, C6×Dic6, C2×C324Q8, He33Q8, C2×C32⋊C12, C2×C4×He3, C2×He33Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C2×Dic6, C6×Q8, C32⋊C6, C3×Dic6, S3×C2×C6, C2×C32⋊C6, C6×Dic6, He33Q8, C22×C32⋊C6, C2×He33Q8

Smallest permutation representation of C2×He33Q8
On 144 points
Generators in S144
(1 76)(2 73)(3 74)(4 75)(5 63)(6 64)(7 61)(8 62)(9 112)(10 109)(11 110)(12 111)(13 48)(14 45)(15 46)(16 47)(17 129)(18 130)(19 131)(20 132)(21 51)(22 52)(23 49)(24 50)(25 123)(26 124)(27 121)(28 122)(29 55)(30 56)(31 53)(32 54)(33 69)(34 70)(35 71)(36 72)(37 65)(38 66)(39 67)(40 68)(41 86)(42 87)(43 88)(44 85)(57 93)(58 94)(59 95)(60 96)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(89 125)(90 126)(91 127)(92 128)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 42 35)(6 43 36)(7 44 33)(8 41 34)(9 115 126)(10 116 127)(11 113 128)(12 114 125)(17 137 117)(18 138 118)(19 139 119)(20 140 120)(25 133 143)(26 134 144)(27 135 141)(28 136 142)(29 40 58)(30 37 59)(31 38 60)(32 39 57)(45 52 74)(46 49 75)(47 50 76)(48 51 73)(53 66 96)(54 67 93)(55 68 94)(56 65 95)(61 85 69)(62 86 70)(63 87 71)(64 88 72)(77 92 110)(78 89 111)(79 90 112)(80 91 109)(81 129 101)(82 130 102)(83 131 103)(84 132 104)(97 107 123)(98 108 124)(99 105 121)(100 106 122)
(1 29 7)(2 30 8)(3 31 5)(4 32 6)(9 135 118)(10 136 119)(11 133 120)(12 134 117)(13 37 41)(14 38 42)(15 39 43)(16 40 44)(17 114 144)(18 115 141)(19 116 142)(20 113 143)(21 59 34)(22 60 35)(23 57 36)(24 58 33)(25 140 128)(26 137 125)(27 138 126)(28 139 127)(45 66 87)(46 67 88)(47 68 85)(48 65 86)(49 93 72)(50 94 69)(51 95 70)(52 96 71)(53 63 74)(54 64 75)(55 61 76)(56 62 73)(77 107 132)(78 108 129)(79 105 130)(80 106 131)(81 111 98)(82 112 99)(83 109 100)(84 110 97)(89 124 101)(90 121 102)(91 122 103)(92 123 104)
(13 37 41)(14 38 42)(15 39 43)(16 40 44)(17 114 144)(18 115 141)(19 116 142)(20 113 143)(21 34 59)(22 35 60)(23 36 57)(24 33 58)(25 128 140)(26 125 137)(27 126 138)(28 127 139)(45 66 87)(46 67 88)(47 68 85)(48 65 86)(49 72 93)(50 69 94)(51 70 95)(52 71 96)(77 107 132)(78 108 129)(79 105 130)(80 106 131)(89 101 124)(90 102 121)(91 103 122)(92 104 123)
(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 9 3 11)(2 12 4 10)(5 133 7 135)(6 136 8 134)(13 125 15 127)(14 128 16 126)(17 57 19 59)(18 60 20 58)(21 114 23 116)(22 113 24 115)(25 44 27 42)(26 43 28 41)(29 118 31 120)(30 117 32 119)(33 141 35 143)(34 144 36 142)(37 137 39 139)(38 140 40 138)(45 92 47 90)(46 91 48 89)(49 80 51 78)(50 79 52 77)(53 84 55 82)(54 83 56 81)(61 99 63 97)(62 98 64 100)(65 101 67 103)(66 104 68 102)(69 105 71 107)(70 108 72 106)(73 111 75 109)(74 110 76 112)(85 121 87 123)(86 124 88 122)(93 131 95 129)(94 130 96 132)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D ··· 6I 6J ··· 6R 12A 12B 12C 12D 12E ··· 12T 12U ··· 12AB order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 3 3 6 6 6 2 2 18 18 18 18 2 2 2 3 ··· 3 6 ··· 6 2 2 2 2 6 ··· 6 18 ··· 18

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + - + + - + + + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 Q8 D6 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C6 C3×Dic6 C32⋊C6 C2×C32⋊C6 C2×C32⋊C6 He3⋊3Q8 kernel C2×He3⋊3Q8 He3⋊3Q8 C2×C32⋊C12 C2×C4×He3 C2×C32⋊4Q8 C32⋊4Q8 C2×C3⋊Dic3 C6×C12 C6×C12 C2×He3 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 4 2 1 2 8 4 2 1 2 2 1 2 4 4 4 2 8 1 2 1 4

Matrix representation of C2×He33Q8 in GL10(𝔽13)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12
,
 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12
,
 3 7 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 0 0 0 0 0 8 10 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 0 0 0 5 8 0 0 0 0 0 0 0 0 0 0 6 9 0 0 0 0 0 0 0 0 6 7 0 0 0 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 0 0 0 0 0 0 1 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 1 0 0 0

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[3,6,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[5,5,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0] >;

C2×He33Q8 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_3Q_8
% in TeX

G:=Group("C2xHe3:3Q8");
// GroupNames label

G:=SmallGroup(432,348);
// by ID

G=gap.SmallGroup(432,348);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,4037,1034,14118]);
// Polycyclic

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

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