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G = (C4×C8)⋊12C4order 128 = 27

3rd semidirect product of C4×C8 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: (C4×C8)⋊12C4, C8⋊C413C4, (C2×C4).42C42, C424C4.1C2, C42.295(C2×C4), C2.5(C424C4), C22.20(C8○D4), C2.C42.7C4, C22.48(C2×C42), C4.69(C42⋊C2), C23.249(C22×C4), (C2×C42).986C22, (C22×C8).375C22, C2.11(C82M4(2)), (C22×C4).1602C23, C22.47(C42⋊C2), C22.7C42.39C2, C2.1(C42.7C22), (C2×C4×C8).12C2, (C2×C8).129(C2×C4), (C2×C8⋊C4).22C2, (C2×C4).912(C4○D4), (C22×C4).105(C2×C4), (C2×C4).592(C22×C4), SmallGroup(128,478)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — (C4×C8)⋊12C4
C1C2C22C2×C4C22×C4C2×C42C424C4 — (C4×C8)⋊12C4
C1C22 — (C4×C8)⋊12C4
C1C22×C4 — (C4×C8)⋊12C4
C1C2C2C22×C4 — (C4×C8)⋊12C4

Generators and relations for (C4×C8)⋊12C4
 G = < a,b,c | a4=b8=c4=1, ab=ba, cac-1=ab4, cbc-1=a2b >

Subgroups: 172 in 120 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C42 [×4], C2×C8 [×8], C2×C8 [×8], C22×C4, C22×C4 [×6], C2.C42 [×4], C4×C8 [×4], C8⋊C4 [×4], C2×C42, C2×C42 [×2], C22×C8 [×4], C22.7C42 [×4], C424C4, C2×C4×C8, C2×C8⋊C4, (C4×C8)⋊12C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], C4○D4 [×4], C2×C42, C42⋊C2 [×6], C8○D4 [×4], C424C4, C82M4(2) [×2], C42.7C22 [×4], (C4×C8)⋊12C4

Smallest permutation representation of (C4×C8)⋊12C4
Regular action on 128 points
Generators in S128
(1 31 23 87)(2 32 24 88)(3 25 17 81)(4 26 18 82)(5 27 19 83)(6 28 20 84)(7 29 21 85)(8 30 22 86)(9 38 74 94)(10 39 75 95)(11 40 76 96)(12 33 77 89)(13 34 78 90)(14 35 79 91)(15 36 80 92)(16 37 73 93)(41 105 97 49)(42 106 98 50)(43 107 99 51)(44 108 100 52)(45 109 101 53)(46 110 102 54)(47 111 103 55)(48 112 104 56)(57 125 113 67)(58 126 114 68)(59 127 115 69)(60 128 116 70)(61 121 117 71)(62 122 118 72)(63 123 119 65)(64 124 120 66)
(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)
(1 119 103 95)(2 64 104 40)(3 113 97 89)(4 58 98 34)(5 115 99 91)(6 60 100 36)(7 117 101 93)(8 62 102 38)(9 82 72 106)(10 27 65 51)(11 84 66 108)(12 29 67 53)(13 86 68 110)(14 31 69 55)(15 88 70 112)(16 25 71 49)(17 57 41 33)(18 114 42 90)(19 59 43 35)(20 116 44 92)(21 61 45 37)(22 118 46 94)(23 63 47 39)(24 120 48 96)(26 122 50 74)(28 124 52 76)(30 126 54 78)(32 128 56 80)(73 81 121 105)(75 83 123 107)(77 85 125 109)(79 87 127 111)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4X8A···8P8Q···8X
order12···24···44···44···48···88···8
size11···11···12···24···42···24···4

56 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C2C4C4C4C4○D4C8○D4
kernel(C4×C8)⋊12C4C22.7C42C424C4C2×C4×C8C2×C8⋊C4C2.C42C4×C8C8⋊C4C2×C4C22
# reps14111888816

Matrix representation of (C4×C8)⋊12C4 in GL5(𝔽17)

160000
04000
00400
00001
000160
,
10000
016800
04100
00002
000150
,
130000
021500
0101500
000716
0001610

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

(C4×C8)⋊12C4 in GAP, Magma, Sage, TeX

(C_4\times C_8)\rtimes_{12}C_4
% in TeX

G:=Group("(C4xC8):12C4");
// GroupNames label

G:=SmallGroup(128,478);
// by ID

G=gap.SmallGroup(128,478);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,58,172]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=c^4=1,a*b=b*a,c*a*c^-1=a*b^4,c*b*c^-1=a^2*b>;
// generators/relations

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