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## G = (C2×C28).Q8order 448 = 26·7

### 8th non-split extension by C2×C28 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C2×C28).Q8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C22×C28 — C2×C4.Dic7 — (C2×C28).Q8
 Lower central C7 — C14 — C2×C14 — (C2×C28).Q8
 Upper central C1 — C4 — C22×C4 — C42⋊C2

Generators and relations for (C2×C28).Q8
G = < a,b,c,d | a2=b28=c4=1, d2=ab21c2, ab=ba, cac-1=dad-1=ab14, cbc-1=b15, dbd-1=b13, dcd-1=ab14c-1 >

Subgroups: 276 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C7⋊C8, C2×C28, C2×C28, C22×C14, M4(2)⋊4C4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C2×C4.Dic7, C7×C42⋊C2, (C2×C28).Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, M4(2)⋊4C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, (C2×C28).Q8

Smallest permutation representation of (C2×C28).Q8
On 112 points
Generators in S112
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(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 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)
(2 16)(4 18)(6 20)(8 22)(10 24)(12 26)(14 28)(29 89 43 103)(30 104 44 90)(31 91 45 105)(32 106 46 92)(33 93 47 107)(34 108 48 94)(35 95 49 109)(36 110 50 96)(37 97 51 111)(38 112 52 98)(39 99 53 85)(40 86 54 100)(41 101 55 87)(42 88 56 102)(57 71)(59 73)(61 75)(63 77)(65 79)(67 81)(69 83)
(1 31 58 112 15 45 72 98)(2 44 59 97 16 30 73 111)(3 29 60 110 17 43 74 96)(4 42 61 95 18 56 75 109)(5 55 62 108 19 41 76 94)(6 40 63 93 20 54 77 107)(7 53 64 106 21 39 78 92)(8 38 65 91 22 52 79 105)(9 51 66 104 23 37 80 90)(10 36 67 89 24 50 81 103)(11 49 68 102 25 35 82 88)(12 34 69 87 26 48 83 101)(13 47 70 100 27 33 84 86)(14 32 71 85 28 46 57 99)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A ··· 8H 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28AP order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 1 1 2 2 2 4 4 4 4 2 2 2 28 ··· 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + - + - + image C1 C2 C2 C4 C4 C4 D4 Q8 D7 Dic7 D14 Dic14 C4×D7 D28 C7⋊D4 M4(2)⋊4C4 (C2×C28).Q8 kernel (C2×C28).Q8 C2×C4.Dic7 C7×C42⋊C2 C2×C7⋊C8 C4.Dic7 C7×C22⋊C4 C2×C28 C2×C28 C42⋊C2 C22⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C7 C1 # reps 1 2 1 4 4 4 3 1 3 6 3 6 12 6 12 2 12

Matrix representation of (C2×C28).Q8 in GL4(𝔽113) generated by

 0 1 0 0 1 0 0 0 96 96 1 2 17 17 0 112
,
 0 56 0 0 56 0 0 0 34 34 111 109 7 7 0 2
,
 1 0 0 0 0 112 0 0 96 96 1 2 17 0 112 112
,
 0 0 1 0 17 17 112 111 98 0 0 0 39 39 0 96
G:=sub<GL(4,GF(113))| [0,1,96,17,1,0,96,17,0,0,1,0,0,0,2,112],[0,56,34,7,56,0,34,7,0,0,111,0,0,0,109,2],[1,0,96,17,0,112,96,0,0,0,1,112,0,0,2,112],[0,17,98,39,0,17,0,39,1,112,0,0,0,111,0,96] >;

(C2×C28).Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{28}).Q_8
% in TeX

G:=Group("(C2xC28).Q8");
// GroupNames label

G:=SmallGroup(448,90);
// by ID

G=gap.SmallGroup(448,90);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,184,1123,851,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=a*b^21*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^14,c*b*c^-1=b^15,d*b*d^-1=b^13,d*c*d^-1=a*b^14*c^-1>;
// generators/relations

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