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G = C22×C2.D8order 128 = 27

Direct product of C22 and C2.D8

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

Aliases: C22×C2.D8, C23.62D8, C24.192D4, C23.24Q16, C88(C22×C4), (C22×C8)⋊15C4, C2.2(C22×D8), C4.2(C22×Q8), (C23×C8).14C2, C4.46(C23×C4), C22.68(C2×D8), C2.2(C22×Q16), C4⋊C4.347C23, C23.86(C4⋊C4), (C2×C8).554C23, (C2×C4).184C24, (C22×C4).604D4, C23.839(C2×D4), C22.45(C2×Q16), (C22×C4).102Q8, (C22×C8).533C22, (C23×C4).695C22, C22.131(C22×D4), (C22×C4).1505C23, (C2×C8)⋊36(C2×C4), C4.64(C2×C4⋊C4), (C2×C4).842(C2×D4), C2.23(C22×C4⋊C4), C22.75(C2×C4⋊C4), (C2×C4).238(C2×Q8), (C2×C4).150(C4⋊C4), (C22×C4⋊C4).43C2, (C2×C4⋊C4).902C22, (C2×C4).572(C22×C4), (C22×C4).495(C2×C4), SmallGroup(128,1640)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C22×C2.D8
C1C2C22C2×C4C22×C4C23×C4C23×C8 — C22×C2.D8
C1C2C4 — C22×C2.D8
C1C24C23×C4 — C22×C2.D8
C1C2C2C2×C4 — C22×C2.D8

Generators and relations for C22×C2.D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 460 in 300 conjugacy classes, 220 normal (12 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×34], C8 [×8], C2×C4, C2×C4 [×27], C2×C4 [×32], C23 [×15], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×28], C22×C4 [×14], C22×C4 [×20], C24, C2.D8 [×16], C2×C4⋊C4 [×12], C2×C4⋊C4 [×6], C22×C8 [×14], C23×C4, C23×C4 [×2], C2×C2.D8 [×12], C22×C4⋊C4 [×2], C23×C8, C22×C2.D8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], D8 [×4], Q16 [×4], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2.D8 [×16], C2×C4⋊C4 [×12], C2×D8 [×6], C2×Q16 [×6], C23×C4, C22×D4, C22×Q8, C2×C2.D8 [×12], C22×C4⋊C4, C22×D8, C22×Q16, C22×C2.D8

Smallest permutation representation of C22×C2.D8
Regular action on 128 points
Generators in S128
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 88)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(49 70)(50 71)(51 72)(52 65)(53 66)(54 67)(55 68)(56 69)(73 95)(74 96)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 113)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 121)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(49 124)(50 125)(51 126)(52 127)(53 128)(54 121)(55 122)(56 123)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(73 117)(74 118)(75 119)(76 120)(77 113)(78 114)(79 115)(80 116)(89 102)(90 103)(91 104)(92 97)(93 98)(94 99)(95 100)(96 101)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 86)(18 87)(19 88)(20 81)(21 82)(22 83)(23 84)(24 85)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(49 103)(50 104)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 112)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 121)(96 122)
(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)
(1 116 59 66)(2 115 60 65)(3 114 61 72)(4 113 62 71)(5 120 63 70)(6 119 64 69)(7 118 57 68)(8 117 58 67)(9 97 32 51)(10 104 25 50)(11 103 26 49)(12 102 27 56)(13 101 28 55)(14 100 29 54)(15 99 30 53)(16 98 31 52)(17 73 86 112)(18 80 87 111)(19 79 88 110)(20 78 81 109)(21 77 82 108)(22 76 83 107)(23 75 84 106)(24 74 85 105)(33 127 48 93)(34 126 41 92)(35 125 42 91)(36 124 43 90)(37 123 44 89)(38 122 45 96)(39 121 46 95)(40 128 47 94)

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

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

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

56 conjugacy classes

class 1 2A···2O4A···4H4I···4X8A···8P
order12···24···44···48···8
size11···12···24···42···2

56 irreducible representations

dim1111122222
type+++++-++-
imageC1C2C2C2C4D4Q8D4D8Q16
kernelC22×C2.D8C2×C2.D8C22×C4⋊C4C23×C8C22×C8C22×C4C22×C4C24C23C23
# reps112211634188

Matrix representation of C22×C2.D8 in GL5(𝔽17)

160000
01000
001600
000160
000016
,
10000
016000
00100
000160
000016
,
10000
01000
001600
000160
000016
,
10000
01000
001600
00080
000015
,
160000
01000
001300
000015
00090

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[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],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,8,0,0,0,0,0,15],[16,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,0,9,0,0,0,15,0] >;

C22×C2.D8 in GAP, Magma, Sage, TeX

C_2^2\times C_2.D_8
% in TeX

G:=Group("C2^2xC2.D8");
// GroupNames label

G:=SmallGroup(128,1640);
// by ID

G=gap.SmallGroup(128,1640);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,568,2804,172]);
// Polycyclic

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

׿
×
𝔽