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## G = C42.123D4order 128 = 27

### 105th non-split extension by C42 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 — C2×C4 — C42.123D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.123D4
 Lower central C1 — C2 — C2×C4 — C42.123D4
 Upper central C1 — C23 — C2×C42 — C42.123D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.123D4

Generators and relations for C42.123D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2b-1c-1 >

Subgroups: 236 in 130 conjugacy classes, 64 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], C23, C42 [×4], C42 [×2], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42, C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C22.4Q16 [×4], C4×C4⋊C4, C2×C4⋊C8, C2×C42.C2, C42.123D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C4○D8 [×2], C8⋊C22, C8.C22, C23.67C23, C23.24D4, C23.36D4, D4.Q8 [×2], Q8.Q8 [×2], C42.123D4

Smallest permutation representation of C42.123D4
Regular action on 128 points
Generators in S128
```(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 29 12 34)(2 30 9 35)(3 31 10 36)(4 32 11 33)(5 123 23 106)(6 124 24 107)(7 121 21 108)(8 122 22 105)(13 39 25 52)(14 40 26 49)(15 37 27 50)(16 38 28 51)(17 118 126 109)(18 119 127 110)(19 120 128 111)(20 117 125 112)(41 62 54 71)(42 63 55 72)(43 64 56 69)(44 61 53 70)(45 84 57 67)(46 81 58 68)(47 82 59 65)(48 83 60 66)(73 100 90 113)(74 97 91 114)(75 98 92 115)(76 99 89 116)(77 94 86 103)(78 95 87 104)(79 96 88 101)(80 93 85 102)
(1 82 50 53)(2 83 51 54)(3 84 52 55)(4 81 49 56)(5 95 112 76)(6 96 109 73)(7 93 110 74)(8 94 111 75)(9 66 38 41)(10 67 39 42)(11 68 40 43)(12 65 37 44)(13 63 31 45)(14 64 32 46)(15 61 29 47)(16 62 30 48)(17 113 124 79)(18 114 121 80)(19 115 122 77)(20 116 123 78)(21 102 119 91)(22 103 120 92)(23 104 117 89)(24 101 118 90)(25 72 36 57)(26 69 33 58)(27 70 34 59)(28 71 35 60)(85 127 97 108)(86 128 98 105)(87 125 99 106)(88 126 100 107)
(1 108 3 106)(2 124 4 122)(5 34 7 36)(6 32 8 30)(9 107 11 105)(10 123 12 121)(13 117 15 119)(14 111 16 109)(17 49 19 51)(18 39 20 37)(21 31 23 29)(22 35 24 33)(25 112 27 110)(26 120 28 118)(38 126 40 128)(41 103 43 101)(42 93 44 95)(45 114 47 116)(46 100 48 98)(50 127 52 125)(53 104 55 102)(54 94 56 96)(57 97 59 99)(58 113 60 115)(61 78 63 80)(62 86 64 88)(65 76 67 74)(66 92 68 90)(69 79 71 77)(70 87 72 85)(73 83 75 81)(82 89 84 91)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 C4 D4 Q8 D4 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel C42.123D4 C22.4Q16 C4×C4⋊C4 C2×C4⋊C8 C2×C42.C2 C42.C2 C42 C4⋊C4 C22×C4 C2×C4 C22 C22 C22 # reps 1 4 1 1 1 8 2 4 2 4 8 1 1

Matrix representation of C42.123D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 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
,
 1 16 0 0 0 0 2 16 0 0 0 0 0 0 14 14 0 0 0 0 14 3 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 6 14 0 0 0 0 6 11 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 10 1 0 0 0 0 1 7

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[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],[1,2,0,0,0,0,16,16,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[6,6,0,0,0,0,14,11,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,10,1,0,0,0,0,1,7] >;`

C42.123D4 in GAP, Magma, Sage, TeX

`C_4^2._{123}D_4`
`% in TeX`

`G:=Group("C4^2.123D4");`
`// GroupNames label`

`G:=SmallGroup(128,721);`
`// by ID`

`G=gap.SmallGroup(128,721);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,436,1018,248,2028,124]);`
`// Polycyclic`

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

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