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## G = (C2×D28)⋊13C4order 448 = 26·7

### 9th semidirect product of C2×D28 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C2×D28)⋊13C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C2×C7⋊D4 — C2×C4○D28 — (C2×D28)⋊13C4
 Lower central C7 — C14 — C2×C14 — (C2×D28)⋊13C4
 Upper central C1 — C4 — C22×C4 — C42⋊C2

Generators and relations for (C2×D28)⋊13C4
G = < a,b,c,d | a2=b28=c2=d4=1, ab=ba, dcd-1=ac=ca, dad-1=ab14, cbc=b-1, bd=db >

Subgroups: 820 in 158 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C23⋊C4, C42⋊C2, C42⋊C2, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C23.C23, C4×Dic7, C4⋊Dic7, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C23.1D14, C23.21D14, C7×C42⋊C2, C2×C4○D28, (C2×D28)⋊13C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C23.C23, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C2×D14⋊C4, (C2×D28)⋊13C4

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

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J ··· 4O 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28AP order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 28 28 1 1 2 2 2 4 4 4 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D7 D14 D14 C4×D7 D28 C7⋊D4 C23.C23 (C2×D28)⋊13C4 kernel (C2×D28)⋊13C4 C23.1D14 C23.21D14 C7×C42⋊C2 C2×C4○D28 C2×Dic14 C2×C4×D7 C2×D28 C2×C28 C42⋊C2 C22⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C7 C1 # reps 1 4 1 1 1 2 4 2 4 3 6 3 12 12 12 2 12

Matrix representation of (C2×D28)⋊13C4 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 28 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 28 0
,
 4 0 0 0 0 0 22 22 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 17 0 0 0 0 0 0 17
,
 23 3 0 0 0 0 27 6 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 17 0 0 12 0 0 0 0 0 0 12 0 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 0 28 0 0 0 0 1 0

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0],[4,22,0,0,0,0,0,22,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[23,27,0,0,0,0,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,17,0,0,0,0,0,0,17,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,28,0] >;

(C2×D28)⋊13C4 in GAP, Magma, Sage, TeX

(C_2\times D_{28})\rtimes_{13}C_4
% in TeX

G:=Group("(C2xD28):13C4");
// GroupNames label

G:=SmallGroup(448,540);
// by ID

G=gap.SmallGroup(448,540);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,422,58,1123,438,18822]);
// Polycyclic

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

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