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## G = (C22×C28)⋊C4order 448 = 26·7

### 2nd semidirect product of C22×C28 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C14 — (C22×C28)⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — D4×C14 — C23⋊Dic7 — (C22×C28)⋊C4
 Lower central C7 — C14 — C2×C14 — C22×C14 — (C22×C28)⋊C4
 Upper central C1 — C2 — C22 — C2×D4 — C22.D4

Generators and relations for (C22×C28)⋊C4
G = < a,b,c,d | a2=b2=c28=d4=1, ab=ba, ac=ca, dad-1=abc14, bc=cb, dbd-1=bc14, dcd-1=abc-1 >

Subgroups: 300 in 68 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C4.D4, C22.D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C23.D4, C4.Dic7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, C28.D4, C23⋊Dic7, C7×C22.D4, (C22×C28)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C23.D4, C23.D7, C23⋊Dic7, (C22×C28)⋊C4

Smallest permutation representation of (C22×C28)⋊C4
On 112 points
Generators in S112
(1 83)(2 84)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(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)
(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 87 22 34)(2 54 9 107)(3 85 24 32)(4 52 11 105)(5 111 26 30)(6 50 13 103)(7 109 28 56)(8 48 15 101)(10 46 17 99)(12 44 19 97)(14 42 21 95)(16 40 23 93)(18 38 25 91)(20 36 27 89)(29 74 110 67)(31 72 112 65)(33 70 86 63)(35 68 88 61)(37 66 90 59)(39 64 92 57)(41 62 94 83)(43 60 96 81)(45 58 98 79)(47 84 100 77)(49 82 102 75)(51 80 104 73)(53 78 106 71)(55 76 108 69)

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

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

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

55 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 14A ··· 14I 14J ··· 14O 14P 14Q 14R 28A ··· 28L 28M ··· 28U order 1 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 14 ··· 14 14 ··· 14 14 14 14 28 ··· 28 28 ··· 28 size 1 1 2 4 4 4 4 4 8 56 56 2 2 2 56 56 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4 8 ··· 8

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - - + + image C1 C2 C2 C2 C4 C4 D4 D4 D7 Dic7 Dic7 D14 C7⋊D4 C7⋊D4 C23⋊C4 C23.D4 C23⋊Dic7 (C22×C28)⋊C4 kernel (C22×C28)⋊C4 C28.D4 C23⋊Dic7 C7×C22.D4 C7×C22⋊C4 C22×C28 C2×C28 C22×C14 C22.D4 C22⋊C4 C22×C4 C2×D4 C2×C4 C23 C14 C7 C2 C1 # reps 1 1 1 1 2 2 1 1 3 3 3 3 6 6 1 2 6 12

Matrix representation of (C22×C28)⋊C4 in GL4(𝔽113) generated by

 112 111 0 0 0 1 0 0 0 0 1 2 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 57 1 0 0 0 56 0 0 0 0 2 0 0 0 0 2
,
 0 0 1 0 0 0 0 1 98 83 0 0 15 15 0 0
G:=sub<GL(4,GF(113))| [112,0,0,0,111,1,0,0,0,0,1,0,0,0,2,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[57,0,0,0,1,56,0,0,0,0,2,0,0,0,0,2],[0,0,98,15,0,0,83,15,1,0,0,0,0,1,0,0] >;

(C22×C28)⋊C4 in GAP, Magma, Sage, TeX

(C_2^2\times C_{28})\rtimes C_4
% in TeX

G:=Group("(C2^2xC28):C4");
// GroupNames label

G:=SmallGroup(448,96);
// by ID

G=gap.SmallGroup(448,96);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,219,675,297,1684,18822]);
// Polycyclic

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

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