Copied to
clipboard

?

G = C24.36D14  order 448 = 26·7

36th non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.36D14, C14.312+ (1+4), C22≀C2⋊8D7, C28⋊2D4⋊15C2, C28⋊D4⋊13C2, (C2×D4).88D14, C22⋊C4.3D14, D14⋊D4⋊15C2, D14⋊C4⋊16C22, Dic7⋊D4⋊6C2, (C2×D28)⋊21C22, C24⋊D7⋊10C2, (C2×C28).33C23, C4⋊Dic7⋊28C22, D14.D4⋊15C2, (C2×C14).139C24, Dic7⋊C4⋊13C22, (C4×Dic7)⋊19C22, C23.D7⋊19C22, C2.33(D4⋊6D14), C7⋊1(C22.54C24), (D4×C14).113C22, C23.D14⋊13C2, C23.18D14⋊6C2, (C23×C14).71C22, (C2×Dic7).64C23, (C22×D7).58C23, C22.160(C23×D7), C23.111(C22×D7), (C22×C14).184C23, (C22×Dic7)⋊17C22, (C2×C4×D7)⋊11C22, (C7×C22≀C2)⋊10C2, (C2×C7⋊D4)⋊11C22, (C2×C4).33(C22×D7), (C7×C22⋊C4).4C22, SmallGroup(448,1048)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C24.36D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14.D4 — C24.36D14
Lower central
C7 — C2×C14 — C24.36D14

Subgroups: 1260 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×9], C22, C22 [×22], C7, C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×12], C23 [×2], C23 [×2], C23 [×5], D7 [×2], C14, C14 [×2], C14 [×4], C42, C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×6], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×9], C24, Dic7 [×6], C28 [×3], D14 [×6], C2×C14, C2×C14 [×16], C22≀C2, C22≀C2 [×2], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2 [×2], C4⋊1D4, C4×D7 [×2], D28, C2×Dic7 [×6], C2×Dic7, C7⋊D4 [×8], C2×C28, C2×C28 [×2], C7×D4 [×3], C22×D7 [×2], C22×C14 [×2], C22×C14 [×2], C22×C14 [×3], C22.54C24, C4×Dic7, Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4 [×2], C23.D7, C23.D7 [×6], C7×C22⋊C4, C7×C22⋊C4 [×2], C2×C4×D7 [×2], C2×D28, C22×Dic7, C2×C7⋊D4 [×8], D4×C14, D4×C14 [×2], C23×C14, C23.D14 [×2], D14.D4 [×2], D14⋊D4 [×2], C23.18D14, C28⋊2D4 [×2], Dic7⋊D4 [×2], C28⋊D4, C24⋊D7 [×2], C7×C22≀C2, C24.36D14

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×3], C22×D7 [×7], C22.54C24, C23×D7, D4⋊6D14 [×3], C24.36D14

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Smallest permutation representation
►On 112 points
Generators in S112
(2 106)(4 108)(6 110)(8 112)(10 86)(12 88)(14 90)(16 92)(18 94)(20 96)(22 98)(24 100)(26 102)(28 104)(29 63)(30 44)(31 65)(32 46)(33 67)(34 48)(35 69)(36 50)(37 71)(38 52)(39 73)(40 54)(41 75)(42 56)(43 77)(45 79)(47 81)(49 83)(51 57)(53 59)(55 61)(58 72)(60 74)(62 76)(64 78)(66 80)(68 82)(70 84)

(2 16)(4 18)(6 20)(8 22)(10 24)(12 26)(14 28)(29 77)(30 64)(31 79)(32 66)(33 81)(34 68)(35 83)(36 70)(37 57)(38 72)(39 59)(40 74)(41 61)(42 76)(43 63)(44 78)(45 65)(46 80)(47 67)(48 82)(49 69)(50 84)(51 71)(52 58)(53 73)(54 60)(55 75)(56 62)(86 100)(88 102)(90 104)(92 106)(94 108)(96 110)(98 112)

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

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

Matrix representation ►G ⊆ GL8(𝔽29)

10000000
128000000
00100000
100280000
00001000
00000100
000000280
000000028
,
10000000
01000000
102800000
100280000
00001000
000002800
000000280
00000001
,
280000000
028000000
002800000
000280000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000028000
000002800
000000280
000000028
,
127000000
028000000
028010000
028100000
00000700
000022000
000000025
00000040
,
100270000
001280000
010280000
000280000
000000025
00000040
00000700
000022000

G:=sub<GL(8,GF(29))| [1,1,0,1,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,27,28,28,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,25,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,28,28,28,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,7,0,0,0,0,0,0,4,0,0,0,0,0,0,25,0,0,0] >;
 

61 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D···4I7A7B7C14A···14I14J···14AA14AB14AC14AD28A···28I
order12222222224444···477714···1414···1414141428···28
size11114444282844428···282222···24···48888···8

61 irreducible representations

dim1111111111222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D7D14D14D142+ (1+4)D4⋊6D14
kernelC24.36D14C23.D14D14.D4D14⋊D4C23.18D14C28⋊2D4Dic7⋊D4C28⋊D4C24⋊D7C7×C22≀C2C22≀C2C22⋊C4C2×D4C24C14C2
# reps12221221213993318

In GAP, Magma, Sage, TeX 

C_2^4._{36}D_{14}
 
% in TeX
 
G:=Group("C2^4.36D14");
 
// GroupNames label
 
G:=SmallGroup(448,1048);
 
// by ID
 
G=gap.SmallGroup(448,1048);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*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
 

׿
×
⋊
ℤ
𝔽
○
≀