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## G = C23⋊2Dic14order 448 = 26·7

### 2nd semidirect product of C23 and Dic14 acting via Dic14/C14=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C23⋊2Dic14
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×C23.D7 — C23⋊2Dic14
 Lower central C7 — C2×C14 — C23⋊2Dic14
 Upper central C1 — C22 — C2×C22⋊C4

Generators and relations for C232Dic14
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=d14, ab=ba, dad-1=ac=ca, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 980 in 242 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C23, C23, C14, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C22⋊Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C232Q8, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C22⋊C4, C2×Dic14, C22×Dic7, C22×C28, C23×C14, C22⋊Dic14, C28.48D4, C2×C23.D7, C14×C22⋊C4, C232Dic14
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C24, D14, C22×Q8, 2+ 1+4, Dic14, C22×D7, C232Q8, C2×Dic14, C23×D7, C22×Dic14, D46D14, C232Dic14

Smallest permutation representation of C232Dic14
On 112 points
Generators in S112
(1 15)(2 47)(3 17)(4 49)(5 19)(6 51)(7 21)(8 53)(9 23)(10 55)(11 25)(12 29)(13 27)(14 31)(16 33)(18 35)(20 37)(22 39)(24 41)(26 43)(28 45)(30 44)(32 46)(34 48)(36 50)(38 52)(40 54)(42 56)(57 71)(58 87)(59 73)(60 89)(61 75)(62 91)(63 77)(64 93)(65 79)(66 95)(67 81)(68 97)(69 83)(70 99)(72 101)(74 103)(76 105)(78 107)(80 109)(82 111)(84 85)(86 100)(88 102)(90 104)(92 106)(94 108)(96 110)(98 112)
(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 29)(27 30)(28 31)(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)
(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 77 15 63)(2 76 16 62)(3 75 17 61)(4 74 18 60)(5 73 19 59)(6 72 20 58)(7 71 21 57)(8 70 22 84)(9 69 23 83)(10 68 24 82)(11 67 25 81)(12 66 26 80)(13 65 27 79)(14 64 28 78)(29 95 43 109)(30 94 44 108)(31 93 45 107)(32 92 46 106)(33 91 47 105)(34 90 48 104)(35 89 49 103)(36 88 50 102)(37 87 51 101)(38 86 52 100)(39 85 53 99)(40 112 54 98)(41 111 55 97)(42 110 56 96)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4L 7A 7B 7C 14A ··· 14U 14V ··· 14AG 28A ··· 28X order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 4 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - + + + + - + image C1 C2 C2 C2 C2 Q8 D7 D14 D14 D14 Dic14 2+ 1+4 D4⋊6D14 kernel C23⋊2Dic14 C22⋊Dic14 C28.48D4 C2×C23.D7 C14×C22⋊C4 C22×C14 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C23 C14 C2 # reps 1 8 4 2 1 4 3 12 6 3 24 2 12

Matrix representation of C232Dic14 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 15 1 0 0 0 0 0 0 28 0 0 0 22 0 0 1
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 22 0 28 0 0 0 7 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 0 1 0 0 0 0 28 0 0 0 0 0 0 0 2 8 0 0 0 0 22 27 0 0 0 0 20 1 0 13 0 0 5 28 16 0
,
 0 17 0 0 0 0 17 0 0 0 0 0 0 0 13 0 12 0 0 0 15 0 26 1 0 0 10 0 16 0 0 0 9 28 13 0

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,15,0,22,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,22,7,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,2,22,20,5,0,0,8,27,1,28,0,0,0,0,0,16,0,0,0,0,13,0],[0,17,0,0,0,0,17,0,0,0,0,0,0,0,13,15,10,9,0,0,0,0,0,28,0,0,12,26,16,13,0,0,0,1,0,0] >;

C232Dic14 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2{\rm Dic}_{14}
% in TeX

G:=Group("C2^3:2Dic14");
// GroupNames label

G:=SmallGroup(448,936);
// by ID

G=gap.SmallGroup(448,936);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,758,675,570,80,18822]);
// Polycyclic

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

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