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## G = C42.3Dic7order 448 = 26·7

### 3rd non-split extension by C42 of Dic7 acting via Dic7/C7=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C42.3Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — Q8×C14 — C28.10D4 — C42.3Dic7
 Lower central C7 — C14 — C2×C14 — C2×C28 — C42.3Dic7
 Upper central C1 — C2 — C22 — C2×Q8 — C4⋊Q8

Generators and relations for C42.3Dic7
G = < a,b,c,d | a4=b4=1, c14=b2, d2=b2c7, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c13 >

Subgroups: 204 in 60 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, M4(2), C2×Q8, C28, C2×C14, C4.10D4, C4⋊Q8, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×Q8, C42.3C4, C4.Dic7, C4×C28, C7×C4⋊C4, Q8×C14, C28.10D4, C7×C4⋊Q8, C42.3Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C42.3C4, C23.D7, C23⋊Dic7, C42.3Dic7

Smallest permutation representation of C42.3Dic7
On 112 points
Generators in S112
(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)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 102 71 88)(58 89 72 103)(59 104 73 90)(60 91 74 105)(61 106 75 92)(62 93 76 107)(63 108 77 94)(64 95 78 109)(65 110 79 96)(66 97 80 111)(67 112 81 98)(68 99 82 85)(69 86 83 100)(70 101 84 87)
(1 43 15 29)(2 30 16 44)(3 45 17 31)(4 32 18 46)(5 47 19 33)(6 34 20 48)(7 49 21 35)(8 36 22 50)(9 51 23 37)(10 38 24 52)(11 53 25 39)(12 40 26 54)(13 55 27 41)(14 42 28 56)(57 88 71 102)(58 103 72 89)(59 90 73 104)(60 105 74 91)(61 92 75 106)(62 107 76 93)(63 94 77 108)(64 109 78 95)(65 96 79 110)(66 111 80 97)(67 98 81 112)(68 85 82 99)(69 100 83 86)(70 87 84 101)
(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 95 22 88 15 109 8 102)(2 108 23 101 16 94 9 87)(3 93 24 86 17 107 10 100)(4 106 25 99 18 92 11 85)(5 91 26 112 19 105 12 98)(6 104 27 97 20 90 13 111)(7 89 28 110 21 103 14 96)(29 78 50 71 43 64 36 57)(30 63 51 84 44 77 37 70)(31 76 52 69 45 62 38 83)(32 61 53 82 46 75 39 68)(33 74 54 67 47 60 40 81)(34 59 55 80 48 73 41 66)(35 72 56 65 49 58 42 79)

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

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

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

55 conjugacy classes

 class 1 2A 2B 4A ··· 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 28A ··· 28R 28S ··· 28AD order 1 2 2 4 ··· 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 4 ··· 4 8 2 2 2 56 56 56 56 2 ··· 2 4 ··· 4 8 ··· 8

55 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + - - + + - image C1 C2 C2 C4 C4 D4 D7 Dic7 Dic7 D14 C7⋊D4 C23⋊C4 C42.3C4 C23⋊Dic7 C42.3Dic7 kernel C42.3Dic7 C28.10D4 C7×C4⋊Q8 C4×C28 Q8×C14 C2×C28 C4⋊Q8 C42 C2×Q8 C2×Q8 C2×C4 C14 C7 C2 C1 # reps 1 2 1 2 2 2 3 3 3 3 12 1 2 6 12

Matrix representation of C42.3Dic7 in GL4(𝔽113) generated by

 112 0 96 48 0 112 14 0 0 0 1 3 0 0 37 112
,
 112 37 49 85 3 1 51 62 0 0 112 110 0 0 76 1
,
 51 8 13 70 109 62 91 29 0 0 33 69 0 0 31 80
,
 97 56 105 91 60 87 57 74 111 19 32 94 94 12 78 10
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,96,14,1,37,48,0,3,112],[112,3,0,0,37,1,0,0,49,51,112,76,85,62,110,1],[51,109,0,0,8,62,0,0,13,91,33,31,70,29,69,80],[97,60,111,94,56,87,19,12,105,57,32,78,91,74,94,10] >;

C42.3Dic7 in GAP, Magma, Sage, TeX

C_4^2._3{\rm Dic}_7
% in TeX

G:=Group("C4^2.3Dic7");
// GroupNames label

G:=SmallGroup(448,105);
// by ID

G=gap.SmallGroup(448,105);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,219,184,1571,570,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^13>;
// generators/relations

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