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G = C2×C42.30C22order 128 = 27

Direct product of C2 and C42.30C22

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

Aliases: C2×C42.30C22, C42.238D4, C42.364C23, C4⋊C4.87C23, (C2×C8).453C23, (C2×C4).332C24, (C22×C4).458D4, C23.875(C2×D4), C4⋊Q8.273C22, (C2×Q8).88C23, C4.23(C4.4D4), C8⋊C4.166C22, (C2×C42).845C22, (C22×C8).459C22, C22.592(C22×D4), (C22×C4).1554C23, Q8⋊C4.201C22, C22.85(C4.4D4), C42.C2.110C22, (C22×Q8).300C22, C22.113(C8.C22), C4.41(C2×C4○D4), (C2×C4⋊Q8).46C2, (C2×C4).512(C2×D4), (C2×C8⋊C4).40C2, C2.43(C2×C4.4D4), C2.38(C2×C8.C22), (C2×C4).711(C4○D4), (C2×C4⋊C4).623C22, (C2×Q8⋊C4).37C2, (C2×C42.C2).32C2, SmallGroup(128,1866)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C42.30C22
C1C2C4C2×C4C22×C4C22×C8C2×C8⋊C4 — C2×C42.30C22
C1C2C2×C4 — C2×C42.30C22
C1C23C2×C42 — C2×C42.30C22
C1C2C2C2×C4 — C2×C42.30C22

Generators and relations for C2×C42.30C22
 G = < a,b,c,d,e | a2=b4=c4=1, d2=c2, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, ebe-1=bc2, dcd-1=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 324 in 192 conjugacy classes, 100 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], Q8 [×12], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×18], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C8⋊C4 [×4], Q8⋊C4 [×16], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42.C2 [×4], C42.C2 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8 [×2], C2×C8⋊C4, C2×Q8⋊C4 [×4], C42.30C22 [×8], C2×C42.C2, C2×C4⋊Q8, C2×C42.30C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8.C22 [×4], C22×D4, C2×C4○D4 [×2], C42.30C22 [×4], C2×C4.4D4, C2×C8.C22 [×2], C2×C42.30C22

Smallest permutation representation of C2×C42.30C22
Regular action on 128 points
Generators in S128
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 79)(34 80)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 95)(42 96)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(65 84)(66 85)(67 86)(68 87)(69 88)(70 81)(71 82)(72 83)(97 116)(98 117)(99 118)(100 119)(101 120)(102 113)(103 114)(104 115)(105 124)(106 125)(107 126)(108 127)(109 128)(110 121)(111 122)(112 123)
(1 60 21 34)(2 57 22 39)(3 62 23 36)(4 59 24 33)(5 64 17 38)(6 61 18 35)(7 58 19 40)(8 63 20 37)(9 26 75 52)(10 31 76 49)(11 28 77 54)(12 25 78 51)(13 30 79 56)(14 27 80 53)(15 32 73 50)(16 29 74 55)(41 113 107 66)(42 118 108 71)(43 115 109 68)(44 120 110 65)(45 117 111 70)(46 114 112 67)(47 119 105 72)(48 116 106 69)(81 91 98 122)(82 96 99 127)(83 93 100 124)(84 90 101 121)(85 95 102 126)(86 92 103 123)(87 89 104 128)(88 94 97 125)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 115 5 119)(2 71 6 67)(3 113 7 117)(4 69 8 65)(9 121 13 125)(10 93 14 89)(11 127 15 123)(12 91 16 95)(17 72 21 68)(18 114 22 118)(19 70 23 66)(20 120 24 116)(25 81 29 85)(26 101 30 97)(27 87 31 83)(28 99 32 103)(33 48 37 44)(34 109 38 105)(35 46 39 42)(36 107 40 111)(41 58 45 62)(43 64 47 60)(49 100 53 104)(50 86 54 82)(51 98 55 102)(52 84 56 88)(57 108 61 112)(59 106 63 110)(73 92 77 96)(74 126 78 122)(75 90 79 94)(76 124 80 128)
(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)| (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,79)(34,80)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,95)(42,96)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(65,84)(66,85)(67,86)(68,87)(69,88)(70,81)(71,82)(72,83)(97,116)(98,117)(99,118)(100,119)(101,120)(102,113)(103,114)(104,115)(105,124)(106,125)(107,126)(108,127)(109,128)(110,121)(111,122)(112,123), (1,60,21,34)(2,57,22,39)(3,62,23,36)(4,59,24,33)(5,64,17,38)(6,61,18,35)(7,58,19,40)(8,63,20,37)(9,26,75,52)(10,31,76,49)(11,28,77,54)(12,25,78,51)(13,30,79,56)(14,27,80,53)(15,32,73,50)(16,29,74,55)(41,113,107,66)(42,118,108,71)(43,115,109,68)(44,120,110,65)(45,117,111,70)(46,114,112,67)(47,119,105,72)(48,116,106,69)(81,91,98,122)(82,96,99,127)(83,93,100,124)(84,90,101,121)(85,95,102,126)(86,92,103,123)(87,89,104,128)(88,94,97,125), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,115,5,119)(2,71,6,67)(3,113,7,117)(4,69,8,65)(9,121,13,125)(10,93,14,89)(11,127,15,123)(12,91,16,95)(17,72,21,68)(18,114,22,118)(19,70,23,66)(20,120,24,116)(25,81,29,85)(26,101,30,97)(27,87,31,83)(28,99,32,103)(33,48,37,44)(34,109,38,105)(35,46,39,42)(36,107,40,111)(41,58,45,62)(43,64,47,60)(49,100,53,104)(50,86,54,82)(51,98,55,102)(52,84,56,88)(57,108,61,112)(59,106,63,110)(73,92,77,96)(74,126,78,122)(75,90,79,94)(76,124,80,128), (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( (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,79)(34,80)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,95)(42,96)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(65,84)(66,85)(67,86)(68,87)(69,88)(70,81)(71,82)(72,83)(97,116)(98,117)(99,118)(100,119)(101,120)(102,113)(103,114)(104,115)(105,124)(106,125)(107,126)(108,127)(109,128)(110,121)(111,122)(112,123), (1,60,21,34)(2,57,22,39)(3,62,23,36)(4,59,24,33)(5,64,17,38)(6,61,18,35)(7,58,19,40)(8,63,20,37)(9,26,75,52)(10,31,76,49)(11,28,77,54)(12,25,78,51)(13,30,79,56)(14,27,80,53)(15,32,73,50)(16,29,74,55)(41,113,107,66)(42,118,108,71)(43,115,109,68)(44,120,110,65)(45,117,111,70)(46,114,112,67)(47,119,105,72)(48,116,106,69)(81,91,98,122)(82,96,99,127)(83,93,100,124)(84,90,101,121)(85,95,102,126)(86,92,103,123)(87,89,104,128)(88,94,97,125), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,115,5,119)(2,71,6,67)(3,113,7,117)(4,69,8,65)(9,121,13,125)(10,93,14,89)(11,127,15,123)(12,91,16,95)(17,72,21,68)(18,114,22,118)(19,70,23,66)(20,120,24,116)(25,81,29,85)(26,101,30,97)(27,87,31,83)(28,99,32,103)(33,48,37,44)(34,109,38,105)(35,46,39,42)(36,107,40,111)(41,58,45,62)(43,64,47,60)(49,100,53,104)(50,86,54,82)(51,98,55,102)(52,84,56,88)(57,108,61,112)(59,106,63,110)(73,92,77,96)(74,126,78,122)(75,90,79,94)(76,124,80,128), (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([(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,79),(34,80),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,95),(42,96),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(65,84),(66,85),(67,86),(68,87),(69,88),(70,81),(71,82),(72,83),(97,116),(98,117),(99,118),(100,119),(101,120),(102,113),(103,114),(104,115),(105,124),(106,125),(107,126),(108,127),(109,128),(110,121),(111,122),(112,123)], [(1,60,21,34),(2,57,22,39),(3,62,23,36),(4,59,24,33),(5,64,17,38),(6,61,18,35),(7,58,19,40),(8,63,20,37),(9,26,75,52),(10,31,76,49),(11,28,77,54),(12,25,78,51),(13,30,79,56),(14,27,80,53),(15,32,73,50),(16,29,74,55),(41,113,107,66),(42,118,108,71),(43,115,109,68),(44,120,110,65),(45,117,111,70),(46,114,112,67),(47,119,105,72),(48,116,106,69),(81,91,98,122),(82,96,99,127),(83,93,100,124),(84,90,101,121),(85,95,102,126),(86,92,103,123),(87,89,104,128),(88,94,97,125)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128)], [(1,115,5,119),(2,71,6,67),(3,113,7,117),(4,69,8,65),(9,121,13,125),(10,93,14,89),(11,127,15,123),(12,91,16,95),(17,72,21,68),(18,114,22,118),(19,70,23,66),(20,120,24,116),(25,81,29,85),(26,101,30,97),(27,87,31,83),(28,99,32,103),(33,48,37,44),(34,109,38,105),(35,46,39,42),(36,107,40,111),(41,58,45,62),(43,64,47,60),(49,100,53,104),(50,86,54,82),(51,98,55,102),(52,84,56,88),(57,108,61,112),(59,106,63,110),(73,92,77,96),(74,126,78,122),(75,90,79,94),(76,124,80,128)], [(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)])

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim1111112224
type++++++++-
imageC1C2C2C2C2C2D4D4C4○D4C8.C22
kernelC2×C42.30C22C2×C8⋊C4C2×Q8⋊C4C42.30C22C2×C42.C2C2×C4⋊Q8C42C22×C4C2×C4C22
# reps1148112284

Matrix representation of C2×C42.30C22 in GL8(𝔽17)

10000000
01000000
001600000
000160000
000016000
000001600
000000160
000000016
,
013000000
130000000
001680000
00410000
0000117016
0000291410
00001565
0000154138
,
160000000
016000000
00100000
00010000
000016200
000016100
00008740
000061613
,
04000000
130000000
001020000
001070000
000014625
0000112164
0000931411
00005074
,
40000000
04000000
00420000
001130000
000016002
000060614
000061207
000016101

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,4,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,11,2,1,15,0,0,0,0,7,9,5,4,0,0,0,0,0,14,6,13,0,0,0,0,16,10,5,8],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,8,6,0,0,0,0,2,1,7,1,0,0,0,0,0,0,4,6,0,0,0,0,0,0,0,13],[0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,14,11,9,5,0,0,0,0,6,2,3,0,0,0,0,0,2,16,14,7,0,0,0,0,5,4,11,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,16,6,6,16,0,0,0,0,0,0,12,1,0,0,0,0,0,6,0,0,0,0,0,0,2,14,7,1] >;

C2×C42.30C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{30}C_2^2
% in TeX

G:=Group("C2xC4^2.30C2^2");
// GroupNames label

G:=SmallGroup(128,1866);
// by ID

G=gap.SmallGroup(128,1866);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,680,758,723,100,2804,172,4037,124]);
// Polycyclic

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

׿
×
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