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## G = (C2×C8).24Q8order 128 = 27

### 24th non-split extension by C2×C8 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).24Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C2.D8 — (C2×C8).24Q8
 Lower central C1 — C2 — C22×C4 — (C2×C8).24Q8
 Upper central C1 — C23 — C2×C42 — (C2×C8).24Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).24Q8

Generators and relations for (C2×C8).24Q8
G = < a,b,c,d | a2=b8=c4=1, d2=ab4c2, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=b3, dcd-1=ab4c-1 >

Subgroups: 216 in 107 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C428C4, C23.65C23, C2×C4.Q8, C2×C2.D8, (C2×C8).24Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C4○D8, C8⋊C22, C8.C22, C23.81C23, D4.2D4, Q8.D4, C23.19D4, C23.20D4, C8.5Q8, C8⋊Q8, (C2×C8).24Q8

Smallest permutation representation of (C2×C8).24Q8
Regular action on 128 points
Generators in S128
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 79)(18 80)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(41 104)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 109)(66 110)(67 111)(68 112)(69 105)(70 106)(71 107)(72 108)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)
(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 58 12 106)(2 117 13 65)(3 64 14 112)(4 115 15 71)(5 62 16 110)(6 113 9 69)(7 60 10 108)(8 119 11 67)(17 88 47 29)(18 92 48 37)(19 86 41 27)(20 90 42 35)(21 84 43 25)(22 96 44 33)(23 82 45 31)(24 94 46 39)(26 78 85 101)(28 76 87 99)(30 74 81 97)(32 80 83 103)(34 79 89 102)(36 77 91 100)(38 75 93 98)(40 73 95 104)(49 111 126 63)(50 70 127 114)(51 109 128 61)(52 68 121 120)(53 107 122 59)(54 66 123 118)(55 105 124 57)(56 72 125 116)
(1 97 54 24)(2 100 55 19)(3 103 56 22)(4 98 49 17)(5 101 50 20)(6 104 51 23)(7 99 52 18)(8 102 53 21)(9 73 128 45)(10 76 121 48)(11 79 122 43)(12 74 123 46)(13 77 124 41)(14 80 125 44)(15 75 126 47)(16 78 127 42)(25 59 89 67)(26 62 90 70)(27 57 91 65)(28 60 92 68)(29 63 93 71)(30 58 94 66)(31 61 95 69)(32 64 96 72)(33 116 83 112)(34 119 84 107)(35 114 85 110)(36 117 86 105)(37 120 87 108)(38 115 88 111)(39 118 81 106)(40 113 82 109)

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

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

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

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 1 2 2 2 2 2 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D4 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel (C2×C8).24Q8 C22.7C42 C22.4Q16 C42⋊8C4 C23.65C23 C2×C4.Q8 C2×C2.D8 C4⋊C4 C2×C8 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 2 1 1 1 1 2 4 2 6 8 1 1

Matrix representation of (C2×C8).24Q8 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 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 1 9 0 0 0 0 13 16 0 0 0 0 0 0 5 12 0 0 0 0 5 5
,
 0 4 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 11 0 0 0 0 11 13
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 16 8 0 0 0 0 0 1 0 0 0 0 0 0 10 1 0 0 0 0 1 7

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

(C2×C8).24Q8 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{24}Q_8
% in TeX

G:=Group("(C2xC8).24Q8");
// GroupNames label

G:=SmallGroup(128,817);
// by ID

G=gap.SmallGroup(128,817);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,64,422,387,58,2028,1027,124]);
// Polycyclic

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

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