Q8xC2^4

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

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

Generators and relations
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 >

Smallest permutation representation
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;

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

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

G = Q₈×C₂⁴ order 128 = 2⁷

Direct product of C₂⁴ and Q₈

G = Q8×C24 order 128 = 27

Direct product of C24 and Q8

G = Q₈×C₂⁴ order 128 = 2⁷

Direct product of C₂⁴ and Q₈