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## G = (C2×C4).28D8order 128 = 27

### 21st non-split extension by C2×C4 of D8 acting via D8/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4).28D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22.4Q16 — (C2×C4).28D8
 Lower central C1 — C2 — C22×C4 — (C2×C4).28D8
 Upper central C1 — C23 — C2×C42 — (C2×C4).28D8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4).28D8

Generators and relations for (C2×C4).28D8
G = < a,b,c,d | a2=b4=c8=1, d2=ab2, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=ac-1 >

Subgroups: 224 in 107 conjugacy classes, 50 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C22×C8 [×2], C22.7C42, C22.4Q16 [×4], C429C4, C23.65C23, (C2×C4).28D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D8 [×2], SD16 [×2], C2×D4, C2×Q8, C4○D4 [×5], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], C2×D8, C2×SD16, C8.C22 [×2], C23.83C23, D4⋊Q8, Q8⋊Q8, C22.D8, C23.47D4, C4.4D8, C42.30C22, (C2×C4).28D8

Smallest permutation representation of (C2×C4).28D8
Regular action on 128 points
Generators in S128
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 80)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)(89 126)(90 127)(91 128)(92 121)(93 122)(94 123)(95 124)(96 125)(105 114)(106 115)(107 116)(108 117)(109 118)(110 119)(111 120)(112 113)
(1 94 87 98)(2 51 88 124)(3 96 81 100)(4 53 82 126)(5 90 83 102)(6 55 84 128)(7 92 85 104)(8 49 86 122)(9 76 41 112)(10 114 42 70)(11 78 43 106)(12 116 44 72)(13 80 45 108)(14 118 46 66)(15 74 47 110)(16 120 48 68)(17 73 37 109)(18 119 38 67)(19 75 39 111)(20 113 40 69)(21 77 33 105)(22 115 34 71)(23 79 35 107)(24 117 36 65)(25 127 63 54)(26 103 64 91)(27 121 57 56)(28 97 58 93)(29 123 59 50)(30 99 60 95)(31 125 61 52)(32 101 62 89)
(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 37 59 46)(2 13 60 24)(3 35 61 44)(4 11 62 22)(5 33 63 42)(6 9 64 20)(7 39 57 48)(8 15 58 18)(10 83 21 25)(12 81 23 31)(14 87 17 29)(16 85 19 27)(26 40 84 41)(28 38 86 47)(30 36 88 45)(32 34 82 43)(49 110 93 67)(50 118 94 73)(51 108 95 65)(52 116 96 79)(53 106 89 71)(54 114 90 77)(55 112 91 69)(56 120 92 75)(66 98 109 123)(68 104 111 121)(70 102 105 127)(72 100 107 125)(74 97 119 122)(76 103 113 128)(78 101 115 126)(80 99 117 124)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + - + + - image C1 C2 C2 C2 C2 Q8 D4 D8 SD16 C4○D4 C8.C22 kernel (C2×C4).28D8 C22.7C42 C22.4Q16 C42⋊9C4 C23.65C23 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 1 4 1 1 2 2 4 4 10 2

Matrix representation of (C2×C4).28D8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 3 12 0 0 0 0 5 14 0 0 0 0 0 0 1 9 0 0 0 0 13 16 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 14 12 0 0 0 0 5 3 0 0 0 0 0 0 16 8 0 0 0 0 0 1 0 0 0 0 0 0 12 5 0 0 0 0 12 12
,
 5 3 0 0 0 0 14 12 0 0 0 0 0 0 4 2 0 0 0 0 0 13 0 0 0 0 0 0 14 3 0 0 0 0 3 3

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

(C2×C4).28D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{28}D_8
% in TeX

G:=Group("(C2xC4).28D8");
// GroupNames label

G:=SmallGroup(128,831);
// by ID

G=gap.SmallGroup(128,831);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,58,2028,1027,124]);
// Polycyclic

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

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