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## G = (C2×C4).D28order 448 = 26·7

### 3rd non-split extension by C2×C4 of D28 acting via D28/C7=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (C2×C4).D28
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — Q8×C14 — C28.23D4 — (C2×C4).D28
 Lower central C7 — C14 — C2×C14 — C2×C28 — (C2×C4).D28
 Upper central C1 — C2 — C22 — C2×Q8 — C4.10D4

Generators and relations for (C2×C4).D28
G = < a,b,c,d | a2=b28=c2=1, d4=b14, ab=ba, ac=ca, dad-1=ab14, cbc=b-1, dbd-1=ab15, dcd-1=b7c >

Subgroups: 428 in 64 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C4.10D4, C4.10D4, C4.4D4, C7⋊C8, C56, D28, C2×Dic7, C2×C28, C7×Q8, C22×D7, C42.C4, C4.Dic7, C4×Dic7, D14⋊C4, C7×M4(2), C2×D28, Q8×C14, C28.10D4, C7×C4.10D4, C28.23D4, (C2×C4).D28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C42.C4, D14⋊C4, C23.1D14, (C2×C4).D28

Smallest permutation representation of (C2×C4).D28
On 112 points
Generators in S112
(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)
(1 102 38 66 15 88 52 80)(2 89 53 67 16 103 39 81)(3 104 40 68 17 90 54 82)(4 91 55 69 18 105 41 83)(5 106 42 70 19 92 56 84)(6 93 29 71 20 107 43 57)(7 108 44 72 21 94 30 58)(8 95 31 73 22 109 45 59)(9 110 46 74 23 96 32 60)(10 97 33 75 24 111 47 61)(11 112 48 76 25 98 34 62)(12 99 35 77 26 85 49 63)(13 86 50 78 27 100 36 64)(14 101 37 79 28 87 51 65)

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

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

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

46 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 56 4 4 4 28 28 2 2 2 8 8 56 56 2 2 2 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D7 D14 C4×D7 D28 C7⋊D4 C23⋊C4 C42.C4 C23.1D14 (C2×C4).D28 kernel (C2×C4).D28 C28.10D4 C7×C4.10D4 C28.23D4 C4×Dic7 C2×D28 C2×C28 C4.10D4 C2×Q8 C2×C4 C2×C4 C2×C4 C14 C7 C2 C1 # reps 1 1 1 1 2 2 2 3 3 6 6 6 1 2 6 3

Matrix representation of (C2×C4).D28 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 22 0 0 112
,
 112 24 0 0 0 0 89 10 0 0 0 0 0 0 15 0 0 0 0 0 43 98 0 0 0 0 0 0 15 0 0 0 104 0 34 98
,
 112 24 0 0 0 0 0 1 0 0 0 0 0 0 107 62 0 0 0 0 45 6 0 0 0 0 60 0 37 87 0 0 2 4 70 76
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 102 0 102 1 0 0 90 87 0 0 0 0 89 76 11 0

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,22,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,89,0,0,0,0,24,10,0,0,0,0,0,0,15,43,0,104,0,0,0,98,0,0,0,0,0,0,15,34,0,0,0,0,0,98],[112,0,0,0,0,0,24,1,0,0,0,0,0,0,107,45,60,2,0,0,62,6,0,4,0,0,0,0,37,70,0,0,0,0,87,76],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,102,90,89,0,0,0,0,87,76,0,0,1,102,0,11,0,0,0,1,0,0] >;

(C2×C4).D28 in GAP, Magma, Sage, TeX

(C_2\times C_4).D_{28}
% in TeX

G:=Group("(C2xC4).D28");
// GroupNames label

G:=SmallGroup(448,34);
// by ID

G=gap.SmallGroup(448,34);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,184,1123,794,297,136,851,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^2=1,d^4=b^14,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^14,c*b*c=b^-1,d*b*d^-1=a*b^15,d*c*d^-1=b^7*c>;
// generators/relations

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