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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

Aliases: (C2×C4)⋊6Q16, (C2×Q16)⋊7C4, C4.18(C4×D4), (C2×C8).244D4, C2.12(C4×Q16), C4.82(C4⋊D4), C8.19(C22⋊C4), C2.3(C4⋊Q16), C4.2(C4.4D4), C22.182(C4×D4), (C22×C4).559D4, C23.802(C2×D4), (C22×Q16).2C2, C22.40(C2×Q16), C2.5(C8.18D4), C2.4(C8.12D4), C22.76(C4○D8), C22.38(C41D4), (C22×C8).493C22, (C22×Q8).39C22, C22.144(C4⋊D4), (C2×C42).1076C22, (C22×C4).1409C23, C23.67C23.10C2, C2.21(C24.3C22), (C2×C4×C8).36C2, (C2×C8).172(C2×C4), C4.37(C2×C22⋊C4), (C2×Q8).95(C2×C4), (C2×C2.D8).11C2, (C2×C4).1354(C2×D4), (C2×C4⋊C4).90C22, (C2×Q8⋊C4).8C2, (C2×C4).600(C4○D4), (C2×C4).423(C22×C4), SmallGroup(128,701)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×C4)⋊6Q16
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — (C2×C4)⋊6Q16
C1C2C2×C4 — (C2×C4)⋊6Q16
C1C23C2×C42 — (C2×C4)⋊6Q16
C1C2C2C22×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 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], Q8 [×12], C23, C42 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C2.C42 [×4], C4×C8 [×2], Q8⋊C4 [×4], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×2], C22×C8 [×2], C2×Q16 [×4], C2×Q16 [×4], C22×Q8 [×2], C23.67C23 [×2], C2×C4×C8, C2×Q8⋊C4 [×2], C2×C2.D8, C22×Q16, (C2×C4)⋊6Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], Q16 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×Q16 [×2], C4○D8 [×2], C24.3C22, C4×Q16 [×2], C8.18D4 [×2], 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)(17 119)(18 120)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 89)(80 90)(81 98)(82 99)(83 100)(84 101)(85 102)(86 103)(87 104)(88 97)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 121)(112 122)
(1 70 37 56)(2 71 38 49)(3 72 39 50)(4 65 40 51)(5 66 33 52)(6 67 34 53)(7 68 35 54)(8 69 36 55)(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 95 112 84)(18 96 105 85)(19 89 106 86)(20 90 107 87)(21 91 108 88)(22 92 109 81)(23 93 110 82)(24 94 111 83)(73 126 97 115)(74 127 98 116)(75 128 99 117)(76 121 100 118)(77 122 101 119)(78 123 102 120)(79 124 103 113)(80 125 104 114)
(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)
(1 110 5 106)(2 109 6 105)(3 108 7 112)(4 107 8 111)(9 114 13 118)(10 113 14 117)(11 120 15 116)(12 119 16 115)(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 98 53 102)(50 97 54 101)(51 104 55 100)(52 103 56 99)(57 85 61 81)(58 84 62 88)(59 83 63 87)(60 82 64 86)(65 80 69 76)(66 79 70 75)(67 78 71 74)(68 77 72 73)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4T8A···8P
order12···24···44···48···8
size11···12···28···82···2

44 irreducible representations

dim111111122222
type++++++++-
imageC1C2C2C2C2C2C4D4D4Q16C4○D4C4○D8
kernel(C2×C4)⋊6Q16C23.67C23C2×C4×C8C2×Q8⋊C4C2×C2.D8C22×Q16C2×Q16C2×C8C22×C4C2×C4C2×C4C22
# reps121211862848

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

160000
01000
00100
000160
000016
,
40000
00100
016000
00040
00004
,
10000
00100
016000
000314
00033
,
10000
0141400
014300
000101
00017

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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