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## G = C4⋊C8⋊14C4order 128 = 27

### 10th 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⋊14C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C4⋊C8⋊14C4
 Lower central C1 — C22 — C4⋊C8⋊14C4
 Upper central C1 — C22×C4 — C4⋊C8⋊14C4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C8⋊14C4

Generators and relations for C4⋊C814C4
G = < a,b,c | a4=b8=c4=1, bab-1=a-1, cac-1=a-1b4, bc=cb >

Subgroups: 188 in 136 conjugacy classes, 88 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C8⋊C4, C8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C22.7C42, C4×C4⋊C4, C2×C4×C8, C2×C8⋊C4, C2×C4⋊C8, C4⋊C814C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C8○D4, C4×C4⋊C4, C4×M4(2), C82M4(2), C89D4, C84Q8, C4⋊C814C4

Smallest permutation representation of C4⋊C814C4
Regular action on 128 points
Generators in S128
(1 114 32 69)(2 70 25 115)(3 116 26 71)(4 72 27 117)(5 118 28 65)(6 66 29 119)(7 120 30 67)(8 68 31 113)(9 36 59 21)(10 22 60 37)(11 38 61 23)(12 24 62 39)(13 40 63 17)(14 18 64 33)(15 34 57 19)(16 20 58 35)(41 77 85 110)(42 111 86 78)(43 79 87 112)(44 105 88 80)(45 73 81 106)(46 107 82 74)(47 75 83 108)(48 109 84 76)(49 103 121 90)(50 91 122 104)(51 97 123 92)(52 93 124 98)(53 99 125 94)(54 95 126 100)(55 101 127 96)(56 89 128 102)
(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 35 96 86)(2 36 89 87)(3 37 90 88)(4 38 91 81)(5 39 92 82)(6 40 93 83)(7 33 94 84)(8 34 95 85)(9 124 79 66)(10 125 80 67)(11 126 73 68)(12 127 74 69)(13 128 75 70)(14 121 76 71)(15 122 77 72)(16 123 78 65)(17 98 47 29)(18 99 48 30)(19 100 41 31)(20 101 42 32)(21 102 43 25)(22 103 44 26)(23 104 45 27)(24 97 46 28)(49 109 116 64)(50 110 117 57)(51 111 118 58)(52 112 119 59)(53 105 120 60)(54 106 113 61)(55 107 114 62)(56 108 115 63)

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

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

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

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 1 1 2 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 Q8 M4(2) C4○D4 C8○D4 kernel C4⋊C8⋊14C4 C22.7C42 C4×C4⋊C4 C2×C4×C8 C2×C8⋊C4 C2×C4⋊C8 C2.C42 C8⋊C4 C4⋊C8 C2×C4⋊C4 C2×C8 C2×C8 C2×C4 C2×C4 C22 # reps 1 2 1 1 2 1 4 8 8 4 2 2 8 4 8

Matrix representation of C4⋊C814C4 in GL6(𝔽17)

 13 12 0 0 0 0 3 4 0 0 0 0 0 0 3 7 0 0 0 0 11 14 0 0 0 0 0 0 7 9 0 0 0 0 2 10
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 4 0 0 0 0 1 0
,
 0 15 0 0 0 0 9 0 0 0 0 0 0 0 0 2 0 0 0 0 9 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13

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

C4⋊C814C4 in GAP, Magma, Sage, TeX

C_4\rtimes C_8\rtimes_{14}C_4
% in TeX

G:=Group("C4:C8:14C4");
// GroupNames label

G:=SmallGroup(128,503);
// by ID

G=gap.SmallGroup(128,503);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,172]);
// Polycyclic

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

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