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## G = Q8×C24order 128 = 27

### Direct product of C24 and Q8

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: Q8×C24, C2.2C26, C4.14C25, C22.17C25, C24.660C23, C23.281C24, C25.100C22, (C24×C4).16C2, (C2×C4).617C24, (C23×C4).715C22, (C22×C4).1593C23, SmallGroup(128,2321)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — Q8×C24
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — Q8×C24
 Lower central C1 — C2 — Q8×C24
 Upper central C1 — C25 — Q8×C24
 Jennings C1 — C2 — Q8×C24

Generators and relations for Q8×C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 3132, all normal (4 characteristic)
C1, C2, C2 [×30], C4 [×48], C22 [×155], C2×C4 [×360], Q8 [×256], C23 [×155], C22×C4 [×420], C2×Q8 [×960], C24 [×31], C23×C4 [×90], C22×Q8 [×560], C25, C24×C4 [×3], Q8×C23 [×60], Q8×C24
Quotients: C1, C2 [×63], C22 [×651], Q8 [×16], C23 [×1395], C2×Q8 [×120], C24 [×651], C22×Q8 [×140], C25 [×63], Q8×C23 [×30], C26, Q8×C24

Smallest permutation representation of Q8×C24
Regular action on 128 points
Generators in S128
(1 9)(2 10)(3 11)(4 12)(5 17)(6 18)(7 19)(8 20)(13 37)(14 38)(15 39)(16 40)(21 55)(22 56)(23 53)(24 54)(25 60)(26 57)(27 58)(28 59)(29 63)(30 64)(31 61)(32 62)(33 47)(34 48)(35 45)(36 46)(41 82)(42 83)(43 84)(44 81)(49 94)(50 95)(51 96)(52 93)(65 77)(66 78)(67 79)(68 80)(69 89)(70 90)(71 91)(72 92)(73 85)(74 86)(75 87)(76 88)(97 116)(98 113)(99 114)(100 115)(101 117)(102 118)(103 119)(104 120)(105 124)(106 121)(107 122)(108 123)(109 125)(110 126)(111 127)(112 128)
(1 63)(2 64)(3 61)(4 62)(5 28)(6 25)(7 26)(8 27)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 59)(18 60)(19 57)(20 58)(21 84)(22 81)(23 82)(24 83)(37 47)(38 48)(39 45)(40 46)(41 53)(42 54)(43 55)(44 56)(49 101)(50 102)(51 103)(52 104)(65 116)(66 113)(67 114)(68 115)(69 105)(70 106)(71 107)(72 108)(73 112)(74 109)(75 110)(76 111)(77 97)(78 98)(79 99)(80 100)(85 128)(86 125)(87 126)(88 127)(89 124)(90 121)(91 122)(92 123)(93 120)(94 117)(95 118)(96 119)
(1 15)(2 16)(3 13)(4 14)(5 41)(6 42)(7 43)(8 44)(9 39)(10 40)(11 37)(12 38)(17 82)(18 83)(19 84)(20 81)(21 57)(22 58)(23 59)(24 60)(25 54)(26 55)(27 56)(28 53)(29 45)(30 46)(31 47)(32 48)(33 61)(34 62)(35 63)(36 64)(49 92)(50 89)(51 90)(52 91)(65 75)(66 76)(67 73)(68 74)(69 95)(70 96)(71 93)(72 94)(77 87)(78 88)(79 85)(80 86)(97 126)(98 127)(99 128)(100 125)(101 123)(102 124)(103 121)(104 122)(105 118)(106 119)(107 120)(108 117)(109 115)(110 116)(111 113)(112 114)
(1 59)(2 60)(3 57)(4 58)(5 29)(6 30)(7 31)(8 32)(9 28)(10 25)(11 26)(12 27)(13 21)(14 22)(15 23)(16 24)(17 63)(18 64)(19 61)(20 62)(33 84)(34 81)(35 82)(36 83)(37 55)(38 56)(39 53)(40 54)(41 45)(42 46)(43 47)(44 48)(49 98)(50 99)(51 100)(52 97)(65 120)(66 117)(67 118)(68 119)(69 112)(70 109)(71 110)(72 111)(73 105)(74 106)(75 107)(76 108)(77 104)(78 101)(79 102)(80 103)(85 124)(86 121)(87 122)(88 123)(89 128)(90 125)(91 126)(92 127)(93 116)(94 113)(95 114)(96 115)
(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)
(1 96 3 94)(2 95 4 93)(5 80 7 78)(6 79 8 77)(9 51 11 49)(10 50 12 52)(13 72 15 70)(14 71 16 69)(17 68 19 66)(18 67 20 65)(21 111 23 109)(22 110 24 112)(25 99 27 97)(26 98 28 100)(29 103 31 101)(30 102 32 104)(33 108 35 106)(34 107 36 105)(37 92 39 90)(38 91 40 89)(41 86 43 88)(42 85 44 87)(45 121 47 123)(46 124 48 122)(53 125 55 127)(54 128 56 126)(57 113 59 115)(58 116 60 114)(61 117 63 119)(62 120 64 118)(73 81 75 83)(74 84 76 82)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2AE 4A ··· 4AV order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 2 type + + + - image C1 C2 C2 Q8 kernel Q8×C24 C24×C4 Q8×C23 C24 # reps 1 3 60 16

Matrix representation of Q8×C24 in GL6(𝔽5)

 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 3 0 0 0 0 1 4
,
 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 4 0 0 0 0 0 2

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,4,2] >;

Q8×C24 in GAP, Magma, Sage, TeX

Q_8\times C_2^4
% in TeX

G:=Group("Q8xC2^4");
// GroupNames label

G:=SmallGroup(128,2321);
// by ID

G=gap.SmallGroup(128,2321);
# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,448,925,456]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=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,b*d=d*b,b*e=e*b,b*f=f*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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