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## G = (C2×Q8).109D4order 128 = 27

### 71st non-split extension by C2×Q8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×Q8).109D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×Q8⋊C4 — (C2×Q8).109D4
 Lower central C1 — C2 — C22×C4 — (C2×Q8).109D4
 Upper central C1 — C23 — C2×C42 — (C2×Q8).109D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×Q8).109D4

Generators and relations for (C2×Q8).109D4
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=ab-1, ebe-1=b-1c, cd=dc, ece-1=b2c, ede-1=b2d-1 >

Subgroups: 232 in 111 conjugacy classes, 46 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], Q8 [×6], C23, C42 [×2], C4⋊C4 [×6], C2×C8 [×6], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×4], Q8⋊C4 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16 [×2], C428C4, C23.67C23, C2×Q8⋊C4 [×2], (C2×Q8).109D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C4○D8 [×2], C8.C22 [×2], C23.11D4, Q8.D4 [×2], C23.20D4 [×2], C42.78C22, C42.30C22, (C2×Q8).109D4

Smallest permutation representation of (C2×Q8).109D4
Regular action on 128 points
Generators in S128
(1 16)(2 13)(3 14)(4 15)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(17 71)(18 72)(19 69)(20 70)(21 27)(22 28)(23 25)(24 26)(37 43)(38 44)(39 41)(40 42)(45 89)(46 90)(47 91)(48 92)(49 55)(50 56)(51 53)(52 54)(57 63)(58 64)(59 61)(60 62)(65 103)(66 104)(67 101)(68 102)(73 79)(74 80)(75 77)(76 78)(81 95)(82 96)(83 93)(84 94)(85 99)(86 100)(87 97)(88 98)(105 111)(106 112)(107 109)(108 110)(113 119)(114 120)(115 117)(116 118)(121 127)(122 128)(123 125)(124 126)
(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 26 3 28)(2 25 4 27)(5 69 7 71)(6 72 8 70)(9 53 11 55)(10 56 12 54)(13 23 15 21)(14 22 16 24)(17 29 19 31)(18 32 20 30)(33 51 35 49)(34 50 36 52)(37 59 39 57)(38 58 40 60)(41 63 43 61)(42 62 44 64)(45 85 47 87)(46 88 48 86)(65 117 67 119)(66 120 68 118)(73 83 75 81)(74 82 76 84)(77 95 79 93)(78 94 80 96)(89 99 91 97)(90 98 92 100)(101 113 103 115)(102 116 104 114)(105 127 107 125)(106 126 108 128)(109 123 111 121)(110 122 112 124)
(1 65 35 78)(2 102 36 75)(3 67 33 80)(4 104 34 73)(5 108 44 85)(6 109 41 98)(7 106 42 87)(8 111 43 100)(9 74 14 101)(10 79 15 66)(11 76 16 103)(12 77 13 68)(17 124 60 89)(18 125 57 48)(19 122 58 91)(20 127 59 46)(21 120 56 93)(22 113 53 82)(23 118 54 95)(24 115 55 84)(25 116 52 81)(26 117 49 94)(27 114 50 83)(28 119 51 96)(29 110 38 99)(30 107 39 88)(31 112 40 97)(32 105 37 86)(45 71 126 62)(47 69 128 64)(61 90 70 121)(63 92 72 123)
(1 100 16 86)(2 91 13 47)(3 98 14 88)(4 89 15 45)(5 83 29 93)(6 76 30 78)(7 81 31 95)(8 74 32 80)(9 107 33 109)(10 126 34 124)(11 105 35 111)(12 128 36 122)(17 77 71 75)(18 96 72 82)(19 79 69 73)(20 94 70 84)(21 87 27 97)(22 46 28 90)(23 85 25 99)(24 48 26 92)(37 67 43 101)(38 120 44 114)(39 65 41 103)(40 118 42 116)(49 123 55 125)(50 112 56 106)(51 121 53 127)(52 110 54 108)(57 119 63 113)(58 66 64 104)(59 117 61 115)(60 68 62 102)

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

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

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

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 1 2 2 2 2 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 C4○D4 C4○D8 C8.C22 kernel (C2×Q8).109D4 C22.7C42 C22.4Q16 C42⋊8C4 C23.67C23 C2×Q8⋊C4 C22×C4 C2×Q8 C2×C4 C22 C22 # reps 1 1 2 1 1 2 2 2 10 8 2

Matrix representation of (C2×Q8).109D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 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 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 10 0 0 0 0 10 16 0 0 0 0 0 0 0 16 0 0 0 0 16 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 2 3 0 0 0 0 4 15 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 14 12 0 0 0 0 5 3
,
 15 5 0 0 0 0 13 2 0 0 0 0 0 0 10 1 0 0 0 0 1 7 0 0 0 0 0 0 5 14 0 0 0 0 3 12

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

(C2×Q8).109D4 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._{109}D_4
% in TeX

G:=Group("(C2xQ8).109D4");
// GroupNames label

G:=SmallGroup(128,806);
// by ID

G=gap.SmallGroup(128,806);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,456,422,387,58,2028,1027,124]);
// Polycyclic

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

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