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## G = C4.68(C4×D4)  order 128 = 27

### 19th non-split extension by C4 of C4×D4 acting via C4×D4/C22⋊C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.68(C4×D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C4×C4⋊C4 — C4.68(C4×D4)
 Lower central C1 — C2 — C2×C4 — C4.68(C4×D4)
 Upper central C1 — C23 — C2×C42 — C4.68(C4×D4)
 Jennings C1 — C2 — C2 — C22×C4 — C4.68(C4×D4)

Generators and relations for C4.68(C4×D4)
G = < a,b,c,d | a4=b4=c4=1, d2=a, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a-1b, dcd-1=a-1c-1 >

Subgroups: 244 in 127 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], Q8 [×6], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], Q8⋊C4 [×4], Q8⋊C4 [×2], C4.Q8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×3], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.67C23, C2×Q8⋊C4 [×2], C2×C4.Q8, C4.68(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, SD16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×SD16, C4○D8, C8.C22 [×2], C24.C22, C4×SD16, Q16⋊C4, D4.D4, Q8.D4, C23.47D4, C23.20D4, C4.68(C4×D4)

Smallest permutation representation of C4.68(C4×D4)
Regular action on 128 points
Generators in S128
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 102 69 40)(2 97 70 35)(3 100 71 38)(4 103 72 33)(5 98 65 36)(6 101 66 39)(7 104 67 34)(8 99 68 37)(9 32 50 82)(10 27 51 85)(11 30 52 88)(12 25 53 83)(13 28 54 86)(14 31 55 81)(15 26 56 84)(16 29 49 87)(17 57 77 111)(18 60 78 106)(19 63 79 109)(20 58 80 112)(21 61 73 107)(22 64 74 110)(23 59 75 105)(24 62 76 108)(41 113 93 127)(42 116 94 122)(43 119 95 125)(44 114 96 128)(45 117 89 123)(46 120 90 126)(47 115 91 121)(48 118 92 124)
(1 13 115 111)(2 106 116 16)(3 11 117 109)(4 112 118 14)(5 9 119 107)(6 110 120 12)(7 15 113 105)(8 108 114 10)(17 102 28 91)(18 94 29 97)(19 100 30 89)(20 92 31 103)(21 98 32 95)(22 90 25 101)(23 104 26 93)(24 96 27 99)(33 80 48 81)(34 84 41 75)(35 78 42 87)(36 82 43 73)(37 76 44 85)(38 88 45 79)(39 74 46 83)(40 86 47 77)(49 70 60 122)(50 125 61 65)(51 68 62 128)(52 123 63 71)(53 66 64 126)(54 121 57 69)(55 72 58 124)(56 127 59 67)
(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)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 SD16 C4○D4 C4○D8 C8.C22 kernel C4.68(C4×D4) C22.7C42 C22.4Q16 C4×C4⋊C4 C23.67C23 C2×Q8⋊C4 C2×C4.Q8 Q8⋊C4 C4⋊C4 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 1 1 1 2 1 8 2 2 4 8 4 2

Matrix representation of C4.68(C4×D4) in GL5(𝔽17)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 13 0 0 0 0 0 4
,
 4 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 4 0
,
 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 16 0
,
 16 0 0 0 0 0 11 4 0 0 0 4 6 0 0 0 0 0 8 0 0 0 0 0 2

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,4],[4,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,16,0],[16,0,0,0,0,0,11,4,0,0,0,4,6,0,0,0,0,0,8,0,0,0,0,0,2] >;

C4.68(C4×D4) in GAP, Magma, Sage, TeX

C_4._{68}(C_4\times D_4)
% in TeX

G:=Group("C4.68(C4xD4)");
// GroupNames label

G:=SmallGroup(128,659);
// by ID

G=gap.SmallGroup(128,659);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,58,2804,718,172]);
// Polycyclic

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

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