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G = C23.Dic14order 448 = 26·7

6th non-split extension by C23 of Dic14 acting via Dic14/C14=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C23.Dic14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C2×C4.Dic7 — C23.Dic14
 Lower central C7 — C14 — C28 — C23.Dic14
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for C23.Dic14
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=bd14, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd27 >

Subgroups: 292 in 102 conjugacy classes, 59 normal (39 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C23, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C28, C2×C14, C2×C14, C8.C4, C2×M4(2), C2×M4(2), C7⋊C8, C7⋊C8, C56, C2×C28, C22×C14, M4(2).C4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C28.53D4, C2×C4.Dic7, C14×M4(2), C23.Dic14
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, Dic14, C4×D7, C7⋊D4, C22×D7, M4(2).C4, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C2×Dic7⋊C4, C23.Dic14

Smallest permutation representation of C23.Dic14
On 112 points
Generators in S112
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 42)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(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 85 15 99 29 57 43 71)(2 112 44 98 30 84 16 70)(3 111 17 69 31 83 45 97)(4 82 46 68 32 110 18 96)(5 81 19 95 33 109 47 67)(6 108 48 94 34 80 20 66)(7 107 21 65 35 79 49 93)(8 78 50 64 36 106 22 92)(9 77 23 91 37 105 51 63)(10 104 52 90 38 76 24 62)(11 103 25 61 39 75 53 89)(12 74 54 60 40 102 26 88)(13 73 27 87 41 101 55 59)(14 100 56 86 42 72 28 58)

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

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

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

82 conjugacy classes

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

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - - + + + - - image C1 C2 C2 C2 C4 D4 Q8 Q8 D7 D14 D14 Dic14 C4×D7 C7⋊D4 Dic14 M4(2).C4 C23.Dic14 kernel C23.Dic14 C28.53D4 C2×C4.Dic7 C14×M4(2) C4.Dic7 C2×C28 C2×C28 C22×C14 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C7 C1 # reps 1 4 2 1 8 2 1 1 3 6 3 6 12 12 6 2 12

Matrix representation of C23.Dic14 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 61 0 112 0 65 0 0 112
,
 1 0 0 0 28 112 0 0 0 0 1 0 65 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 2 16 0 0 109 111 0 0 49 36 0 56 34 68 64 0
,
 44 0 106 0 75 0 15 1 85 0 69 0 90 15 55 0
G:=sub<GL(4,GF(113))| [1,0,61,65,0,1,0,0,0,0,112,0,0,0,0,112],[1,28,0,65,0,112,0,0,0,0,1,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[2,109,49,34,16,111,36,68,0,0,0,64,0,0,56,0],[44,75,85,90,0,0,0,15,106,15,69,55,0,1,0,0] >;

C23.Dic14 in GAP, Magma, Sage, TeX

C_2^3.{\rm Dic}_{14}
% in TeX

G:=Group("C2^3.Dic14");
// GroupNames label

G:=SmallGroup(448,658);
// by ID

G=gap.SmallGroup(448,658);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,422,58,136,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=b*d^14,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^27>;
// generators/relations

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