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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

Aliases: (C2×Q8).109D4, C23.929(C2×D4), (C22×C4).159D4, C428C4.15C2, (C22×C8).87C22, C4.40(C422C2), C22.121(C4○D8), C22.4Q16.26C2, (C2×C42).383C22, C2.22(Q8.D4), (C22×Q8).79C22, C22.250(C4⋊D4), (C22×C4).1463C23, C2.9(C23.11D4), C4.29(C22.D4), C22.96(C4.4D4), C2.10(C23.20D4), C22.139(C8.C22), C22.7C42.14C2, C23.67C23.20C2, C2.7(C42.30C22), C2.10(C42.78C22), C22.119(C22.D4), (C2×C4).1062(C2×D4), (C2×C4).625(C4○D4), (C2×C4⋊C4).148C22, (C2×Q8⋊C4).17C2, SmallGroup(128,806)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×Q8).109D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×Q8⋊C4 — (C2×Q8).109D4
C1C2C22×C4 — (C2×Q8).109D4
C1C23C2×C42 — (C2×Q8).109D4
C1C2C2C22×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···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim11111122224
type++++++++-
imageC1C2C2C2C2C2D4D4C4○D4C4○D8C8.C22
kernel(C2×Q8).109D4C22.7C42C22.4Q16C428C4C23.67C23C2×Q8⋊C4C22×C4C2×Q8C2×C4C22C22
# reps112112221082

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

100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
0011000
00101600
0000016
0000160
,
100000
010000
000100
0016000
0000160
0000016
,
230000
4150000
0013000
0001300
00001412
000053
,
1550000
1320000
0010100
001700
0000514
0000312

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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