C2^3xQ16 - GroupNames

G = C₂³xQ₁₆ order 128 = 2⁷

Direct product of C₂³ and Q₁₆

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂³xQ₁₆, C₄.₃C₂⁵, C₈.₁₉C₂⁴, Q₈.₁C₂⁴, C₂⁴.₁₉₇D₄, (C₂³xC₈).₁₆C₂, C₄.₂₉(C₂²xD₄), C₂.₃₈(D₄xC₂³), (C₂xC₈).₅₇₄C₂³, (C₂xC₄).₆₀₉C₂⁴, C₂³.₈₉₅(C₂xD₄), (C₂²xC₄).₆₂₉D₄, (Q₈xC₂³).₁₅C₂, (C₂xQ₈).₄₇₃C₂³, (C₂²xC₈).₅₄₄C₂², (C₂³xC₄).₇₁₃C₂², C₂².₁₆₆(C₂²xD₄), (C₂²xC₄).₁₅₉₁C₂³, (C₂²xQ₈).₅₀₃C₂², (C₂xC₄).₈₈₂(C₂xD₄), SmallGroup(128,2308)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₄ — C₂³xQ₁₆

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂³xC₄ — Q₈xC₂³ — C₂³xQ₁₆

Lower central

C₁ — C₂ — C₄ — C₂³xQ₁₆

Upper central

C₁ — C₂⁴ — C₂³xC₄ — C₂³xQ₁₆

Jennings

C₁ — C₂ — C₂ — C₄ — C₂³xQ₁₆

Generators and relations for C₂³xQ₁₆
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=d⁴, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 988 in 732 conjugacy classes, 476 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₈, C₂xC₄, C₂xC₄, Q₈, Q₈, C₂³, C₂xC₈, Q₁₆, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₂⁴, C₂²xC₈, C₂xQ₁₆, C₂³xC₄, C₂³xC₄, C₂²xQ₈, C₂²xQ₈, C₂³xC₈, C₂²xQ₁₆, Q₈xC₂³, C₂³xQ₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂xD₄, C₂⁴, C₂xQ₁₆, C₂²xD₄, C₂⁵, C₂²xQ₁₆, D₄xC₂³, C₂³xQ₁₆

Smallest permutation representation of C₂³xQ₁₆
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2O	4A	···	4H	4I	···	4X	8A	···	8P
order	1	2	···	2	4	···	4	4	···	4	8	···	8
size	1	1	···	1	2	···	2	4	···	4	2	···	2

56 irreducible representations

dim	1	1	1	1	2	2	2
type	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	D₄	D₄	Q₁₆
kernel	C₂³xQ₁₆	C₂³xC₈	C₂²xQ₁₆	Q₈xC₂³	C₂²xC₄	C₂⁴	C₂³
# reps	1	1	28	2	7	1	16

Matrix representation of C₂³xQ₁₆ ►in GL₇(F₁₇)

16	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	16	0	0	0
0	0	0	0	16	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

16	0	0	0	0	0	0
0	16	0	0	0	0	0
0	0	16	0	0	0	0
0	0	0	16	0	0	0
0	0	0	0	16	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	16	0	0	0	0	0
0	0	16	0	0	0	0
0	0	0	16	0	0	0
0	0	0	0	16	0	0
0	0	0	0	0	16	0
0	0	0	0	0	0	16

1	0	0	0	0	0	0
0	7	13	0	0	0	0
0	4	10	0	0	0	0
0	0	0	10	4	0	0
0	0	0	13	7	0	0
0	0	0	0	0	3	14
0	0	0	0	0	3	3

1	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	0	16	0	0
0	0	0	16	0	0	0
0	0	0	0	0	10	16
0	0	0	0	0	16	7

G:=sub<GL(7,GF(17))| [16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,7,4,0,0,0,0,0,13,10,0,0,0,0,0,0,0,10,13,0,0,0,0,0,4,7,0,0,0,0,0,0,0,3,3,0,0,0,0,0,14,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,10,16,0,0,0,0,0,16,7] >;

C₂³xQ₁₆ in GAP, Magma, Sage, TeX

C_2^3\times Q_{16}

% in TeX

G:=Group("C2^3xQ16");

// GroupNames label

G:=SmallGroup(128,2308);

// by ID

G=gap.SmallGroup(128,2308);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,456,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C23xQ16 order 128 = 27

Direct product of C23 and Q16

G = C₂³xQ₁₆ order 128 = 2⁷

Direct product of C₂³ and Q₁₆