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## G = C24.4Dic7order 448 = 26·7

### 2nd non-split extension by C24 of Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C24.4Dic7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C28.55D4 — C24.4Dic7
 Lower central C7 — C2×C14 — C24.4Dic7
 Upper central C1 — C2×C4 — C23×C4

Generators and relations for C24.4Dic7
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=d, f2=de7, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 452 in 190 conjugacy classes, 79 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, C2×M4(2), C23×C4, C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C24.4C4, C2×C7⋊C8, C4.Dic7, C22×C28, C22×C28, C22×C28, C23×C14, C28.55D4, C2×C4.Dic7, C23×C28, C24.4Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, M4(2), C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C2×M4(2), C2×Dic7, C7⋊D4, C22×D7, C24.4C4, C4.Dic7, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C4.Dic7, C2×C23.D7, C24.4Dic7

Smallest permutation representation of C24.4Dic7
On 112 points
Generators in S112
(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)
(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 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)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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)
(1 100 22 93 15 86 8 107)(2 85 23 106 16 99 9 92)(3 98 24 91 17 112 10 105)(4 111 25 104 18 97 11 90)(5 96 26 89 19 110 12 103)(6 109 27 102 20 95 13 88)(7 94 28 87 21 108 14 101)(29 63 50 84 43 77 36 70)(30 76 51 69 44 62 37 83)(31 61 52 82 45 75 38 68)(32 74 53 67 46 60 39 81)(33 59 54 80 47 73 40 66)(34 72 55 65 48 58 41 79)(35 57 56 78 49 71 42 64)

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 7A 7B 7C 8A ··· 8H 14A ··· 14AS 28A ··· 28AV order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 2 2 28 ··· 28 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C4 D4 D7 M4(2) Dic7 D14 Dic7 C7⋊D4 C4.Dic7 kernel C24.4Dic7 C28.55D4 C2×C4.Dic7 C23×C28 C22×C28 C23×C14 C2×C28 C23×C4 C2×C14 C22×C4 C22×C4 C24 C2×C4 C22 # reps 1 4 2 1 6 2 4 3 8 9 9 3 24 48

Matrix representation of C24.4Dic7 in GL4(𝔽113) generated by

 1 0 0 0 12 112 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
,
 28 0 0 0 79 109 0 0 0 0 105 0 0 0 0 99
,
 41 12 0 0 86 72 0 0 0 0 0 1 0 0 98 0
G:=sub<GL(4,GF(113))| [1,12,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[28,79,0,0,0,109,0,0,0,0,105,0,0,0,0,99],[41,86,0,0,12,72,0,0,0,0,0,98,0,0,1,0] >;

C24.4Dic7 in GAP, Magma, Sage, TeX

C_2^4._4{\rm Dic}_7
% in TeX

G:=Group("C2^4.4Dic7");
// GroupNames label

G:=SmallGroup(448,741);
// by ID

G=gap.SmallGroup(448,741);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,253,758,136,18822]);
// Polycyclic

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

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