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G = C2.(C8⋊D4)  order 128 = 27

4th central extension by C2 of C8⋊D4

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

Aliases: C4.124(C4×D4), (C2×C8).199D4, Q8⋊C411C4, C2.4(C8⋊D4), C2.3(C8.D4), (C22×C4).130D4, C22.169(C4×D4), C23.786(C2×D4), C4.3(C422C2), C4.9(C42⋊C2), C2.10(Q16⋊C4), C22.87(C8⋊C22), C22.4Q16.48C2, (C22×C8).397C22, (C2×C42).302C22, C2.16(SD16⋊C4), (C22×Q8).31C22, C22.132(C4⋊D4), (C22×C4).1386C23, C22.60(C4.4D4), C4.94(C22.D4), C22.76(C8.C22), C23.65C23.9C2, C23.67C23.9C2, C2.20(C24.C22), C2.2(C42.30C22), C2.2(C42.28C22), C4⋊C4.83(C2×C4), (C2×C8).146(C2×C4), (C2×Q8).85(C2×C4), (C2×C8⋊C4).31C2, (C2×C4).1343(C2×D4), (C2×C4⋊C4).72C22, (C2×C4).581(C4○D4), (C2×C4).404(C22×C4), (C2×Q8⋊C4).33C2, SmallGroup(128,667)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2.(C8⋊D4)
C1C2C22C2×C4C22×C4C22×C8C2×C8⋊C4 — C2.(C8⋊D4)
C1C2C2×C4 — C2.(C8⋊D4)
C1C23C2×C42 — C2.(C8⋊D4)
C1C2C2C22×C4 — C2.(C8⋊D4)

Generators and relations for C2.(C8⋊D4)
 G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab3, dcd-1=c-1 >

Subgroups: 244 in 124 conjugacy classes, 56 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×8], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×20], Q8 [×6], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], C8⋊C4 [×2], Q8⋊C4 [×4], Q8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×Q8, C22.4Q16 [×2], C23.65C23, C23.67C23, C2×C8⋊C4, C2×Q8⋊C4 [×2], C2.(C8⋊D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C8⋊C22, C8.C22 [×3], C24.C22, SD16⋊C4, Q16⋊C4, C8⋊D4, C8.D4, C42.28C22, C42.30C22, C2.(C8⋊D4)

Smallest permutation representation of C2.(C8⋊D4)
Regular action on 128 points
Generators in S128
(1 71)(2 72)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(41 125)(42 126)(43 127)(44 128)(45 121)(46 122)(47 123)(48 124)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 117)(90 118)(91 119)(92 120)(93 113)(94 114)(95 115)(96 116)
(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 33 29 118)(2 15 30 89)(3 39 31 116)(4 13 32 95)(5 37 25 114)(6 11 26 93)(7 35 27 120)(8 9 28 91)(10 49 92 69)(12 55 94 67)(14 53 96 65)(16 51 90 71)(17 83 104 126)(18 106 97 41)(19 81 98 124)(20 112 99 47)(21 87 100 122)(22 110 101 45)(23 85 102 128)(24 108 103 43)(34 50 119 70)(36 56 113 68)(38 54 115 66)(40 52 117 72)(42 59 107 80)(44 57 109 78)(46 63 111 76)(48 61 105 74)(58 84 79 127)(60 82 73 125)(62 88 75 123)(64 86 77 121)
(1 47 71 123)(2 126 72 42)(3 45 65 121)(4 124 66 48)(5 43 67 127)(6 122 68 46)(7 41 69 125)(8 128 70 44)(9 102 34 78)(10 73 35 97)(11 100 36 76)(12 79 37 103)(13 98 38 74)(14 77 39 101)(15 104 40 80)(16 75 33 99)(17 117 59 89)(18 92 60 120)(19 115 61 95)(20 90 62 118)(21 113 63 93)(22 96 64 116)(23 119 57 91)(24 94 58 114)(25 108 55 84)(26 87 56 111)(27 106 49 82)(28 85 50 109)(29 112 51 88)(30 83 52 107)(31 110 53 86)(32 81 54 105)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim111111122244
type+++++++++-
imageC1C2C2C2C2C2C4D4D4C4○D4C8⋊C22C8.C22
kernelC2.(C8⋊D4)C22.4Q16C23.65C23C23.67C23C2×C8⋊C4C2×Q8⋊C4Q8⋊C4C2×C8C22×C4C2×C4C22C22
# reps121112822813

Matrix representation of C2.(C8⋊D4) in GL8(𝔽17)

160000000
016000000
00100000
00010000
000016000
000001600
000000160
000000016
,
1615000000
01000000
001600000
000160000
000001602
00003339
000071211
00001011013
,
130000000
44000000
001040000
001370000
0000815145
000043133
0000610127
0000167711
,
12000000
1616000000
001370000
001040000
000010121613
0000145101
00006679
0000211312

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,7,10,0,0,0,0,16,3,12,1,0,0,0,0,0,3,1,10,0,0,0,0,2,9,1,13],[13,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,10,13,0,0,0,0,0,0,4,7,0,0,0,0,0,0,0,0,8,4,6,16,0,0,0,0,15,3,10,7,0,0,0,0,14,13,12,7,0,0,0,0,5,3,7,11],[1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,13,10,0,0,0,0,0,0,7,4,0,0,0,0,0,0,0,0,10,14,6,2,0,0,0,0,12,5,6,1,0,0,0,0,16,10,7,13,0,0,0,0,13,1,9,12] >;

C2.(C8⋊D4) in GAP, Magma, Sage, TeX

C_2.(C_8\rtimes D_4)
% in TeX

G:=Group("C2.(C8:D4)");
// GroupNames label

G:=SmallGroup(128,667);
// by ID

G=gap.SmallGroup(128,667);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,58,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^3,d*c*d^-1=c^-1>;
// generators/relations

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