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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — (C4×C8)⋊12C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C42⋊4C4 — (C4×C8)⋊12C4
 Lower central C1 — C22 — (C4×C8)⋊12C4
 Upper central C1 — C22×C4 — (C4×C8)⋊12C4
 Jennings C1 — C2 — C2 — C22×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, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C8⋊C4, C2×C42, C2×C42, C22×C8, C22.7C42, C424C4, C2×C4×C8, C2×C8⋊C4, (C4×C8)⋊12C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C4○D4, C2×C42, C42⋊C2, C8○D4, C424C4, C82M4(2), C42.7C22, (C4×C8)⋊12C4

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4○D4 C8○D4 kernel (C4×C8)⋊12C4 C22.7C42 C42⋊4C4 C2×C4×C8 C2×C8⋊C4 C2.C42 C4×C8 C8⋊C4 C2×C4 C22 # reps 1 4 1 1 1 8 8 8 8 16

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

 16 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 16 0
,
 1 0 0 0 0 0 16 8 0 0 0 4 1 0 0 0 0 0 0 2 0 0 0 15 0
,
 13 0 0 0 0 0 2 15 0 0 0 10 15 0 0 0 0 0 7 16 0 0 0 16 10

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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