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## G = (C2×C4).21Q16order 128 = 27

### 18th non-split extension by C2×C4 of Q16 acting via Q16/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4).21Q16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.65C23 — (C2×C4).21Q16
 Lower central C1 — C2 — C22×C4 — (C2×C4).21Q16
 Upper central C1 — C23 — C2×C42 — (C2×C4).21Q16
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4).21Q16

Generators and relations for (C2×C4).21Q16
G = < a,b,c,d | a2=b4=c8=1, d2=b2c4, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=c-1 >

Subgroups: 224 in 112 conjugacy classes, 52 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22 [×3], C22 [×4], C8 [×3], C2×C4 [×2], C2×C4 [×6], C2×C4 [×19], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4⋊C8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16, C22.4Q16 [×2], C23.65C23 [×2], C2×C4⋊C8, C2×C2.D8, (C2×C4).21Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, Q16 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×Q16, C4○D8, C8⋊C22 [×2], C23.81C23, C8.18D4, C82D4, C4.Q16 [×2], D4.Q8 [×2], (C2×C4).21Q16

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

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

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

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

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 2 4 type + + + + + - + + - + image C1 C2 C2 C2 C2 Q8 D4 D4 Q16 C4○D4 C4○D8 C8⋊C22 kernel (C2×C4).21Q16 C22.4Q16 C23.65C23 C2×C4⋊C8 C2×C2.D8 C4⋊C4 C2×C8 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 3 2 1 1 4 2 2 4 6 4 2

Matrix representation of (C2×C4).21Q16 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 2 0 0 0 0 7 11 0 0 0 0 0 0 0 4 0 0 0 0 13 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 8 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 6 11 0 0 0 0 3 0
,
 4 15 0 0 0 0 0 13 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 7 0 0 0 0 5 7

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

(C2×C4).21Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{21}Q_{16}
% in TeX

G:=Group("(C2xC4).21Q16");
// GroupNames label

G:=SmallGroup(128,819);
// by ID

G=gap.SmallGroup(128,819);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,394,718,172]);
// Polycyclic

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

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