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## G = (C2×C4)⋊6Q16order 128 = 27

### 1st semidirect product of C2×C4 and Q16 acting via Q16/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C4)⋊6Q16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — (C2×C4)⋊6Q16
 Lower central C1 — C2 — C2×C4 — (C2×C4)⋊6Q16
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊6Q16
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊6Q16

Generators and relations for (C2×C4)⋊6Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=ab-1, dcd-1=c-1 >

Subgroups: 292 in 156 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C4×C8, Q8⋊C4, C2.D8, C2×C42, C2×C4⋊C4, C22×C8, C2×Q16, C2×Q16, C22×Q8, C23.67C23, C2×C4×C8, C2×Q8⋊C4, C2×C2.D8, C22×Q16, (C2×C4)⋊6Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, Q16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C41D4, C2×Q16, C4○D8, C24.3C22, C4×Q16, C8.18D4, C4⋊Q16, C8.12D4, (C2×C4)⋊6Q16

Smallest permutation representation of (C2×C4)⋊6Q16
Regular action on 128 points
Generators in S128
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 104)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 127)(112 128)
(1 70 37 50)(2 71 38 51)(3 72 39 52)(4 65 40 53)(5 66 33 54)(6 67 34 55)(7 68 35 56)(8 69 36 49)(9 63 26 46)(10 64 27 47)(11 57 28 48)(12 58 29 41)(13 59 30 42)(14 60 31 43)(15 61 32 44)(16 62 25 45)(17 95 106 86)(18 96 107 87)(19 89 108 88)(20 90 109 81)(21 91 110 82)(22 92 111 83)(23 93 112 84)(24 94 105 85)(73 124 103 115)(74 125 104 116)(75 126 97 117)(76 127 98 118)(77 128 99 119)(78 121 100 120)(79 122 101 113)(80 123 102 114)
(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)
(1 112 5 108)(2 111 6 107)(3 110 7 106)(4 109 8 105)(9 116 13 120)(10 115 14 119)(11 114 15 118)(12 113 16 117)(17 39 21 35)(18 38 22 34)(19 37 23 33)(20 36 24 40)(25 126 29 122)(26 125 30 121)(27 124 31 128)(28 123 32 127)(41 95 45 91)(42 94 46 90)(43 93 47 89)(44 92 48 96)(49 100 53 104)(50 99 54 103)(51 98 55 102)(52 97 56 101)(57 87 61 83)(58 86 62 82)(59 85 63 81)(60 84 64 88)(65 74 69 78)(66 73 70 77)(67 80 71 76)(68 79 72 75)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 Q16 C4○D4 C4○D8 kernel (C2×C4)⋊6Q16 C23.67C23 C2×C4×C8 C2×Q8⋊C4 C2×C2.D8 C22×Q16 C2×Q16 C2×C8 C22×C4 C2×C4 C2×C4 C22 # reps 1 2 1 2 1 1 8 6 2 8 4 8

Matrix representation of (C2×C4)⋊6Q16 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 3 14 0 0 0 3 3
,
 1 0 0 0 0 0 14 14 0 0 0 14 3 0 0 0 0 0 10 1 0 0 0 1 7

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,14,14,0,0,0,14,3,0,0,0,0,0,10,1,0,0,0,1,7] >;

(C2×C4)⋊6Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_6Q_{16}
% in TeX

G:=Group("(C2xC4):6Q16");
// GroupNames label

G:=SmallGroup(128,701);
// by ID

G=gap.SmallGroup(128,701);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,2019,248,2804,172]);
// Polycyclic

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

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