C2^4xC8

direct product, p-group, abelian, monomial

Aliases: C₂⁴×C₈, SmallGroup(128,2301)

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

Derived series

C₁ — C₂⁴×C₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂⁴×C₄ — C₂⁴×C₈

Lower central

C₁ — C₂⁴×C₈

Upper central

C₁ — C₂⁴×C₈

Jennings

C₁ — C₂ — C₂ — C₄ — C₂⁴×C₈

Subgroups: 988, all normal (6 characteristic)
C₁, C₂, C₂^[×30], C₄, C₄^[×15], C₂²^[×155], C₈^[×16], C₂×C₄^[×120], C₂³^[×155], C₂×C₈^[×120], C₂²×C₄^[×140], C₂⁴^[×31], C₂²×C₈^[×140], C₂³×C₄^[×30], C₂⁵, C₂³×C₈^[×30], C₂⁴×C₄, C₂⁴×C₈

Quotients:
C₁, C₂^[×31], C₄^[×16], C₂²^[×155], C₈^[×16], C₂×C₄^[×120], C₂³^[×155], C₂×C₈^[×120], C₂²×C₄^[×140], C₂⁴^[×31], C₂²×C₈^[×140], C₂³×C₄^[×30], C₂⁵, C₂³×C₈^[×30], C₂⁴×C₄, C₂⁴×C₈

Generators and relations
G = < a,b,c,d,e | a²=b²=c²=d²=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Smallest permutation representation
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

(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)

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

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

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

Matrix representation ►G ⊆ GL₅(𝔽₁₇)

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

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

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

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

8	0	0	0	0
0	2	0	0	0
0	0	16	0	0
0	0	0	9	0
0	0	0	0	16

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16],[8,0,0,0,0,0,2,0,0,0,0,0,16,0,0,0,0,0,9,0,0,0,0,0,16] >;

128 conjugacy classes

class	1	2A	···	2AE	4A	···	4AF	8A	···	8BL
order	1	2	···	2	4	···	4	8	···	8
size	1	1	···	1	1	···	1	1	···	1

128 irreducible representations

dim	1	1	1	1	1	1
type	+	+	+
image	C₁	C₂	C₂	C₄	C₄	C₈
kernel	C₂⁴×C₈	C₂³×C₈	C₂⁴×C₄	C₂³×C₄	C₂⁵	C₂⁴
# reps	1	30	1	30	2	64

In GAP, Magma, Sage, TeX

C_2^4\times C_8

% in TeX

G:=Group("C2^4xC8");

// GroupNames label

G:=SmallGroup(128,2301);

// by ID

G=gap.SmallGroup(128,2301);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,124]);

// Polycyclic

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

// generators/relations

G = C₂⁴×C₈ order 128 = 2⁷

Abelian group of type [2,2,2,2,8]

G = C24×C8 order 128 = 27

Abelian group of type [2,2,2,2,8]

G = C₂⁴×C₈ order 128 = 2⁷