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G = C42.19C23order 128 = 27

19th non-split extension by C42 of C23 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.19C23, C4:C4.340D4, C4oD4.19D4, D4.10(C2xD4), Q8.10(C2xD4), C2.9(Q8oD8), C4:2Q16:21C2, D4.D4:4C2, C4:C8.41C22, (C2xC8).21C23, C4.77(C22xD4), D4.2D4:17C2, C4:C4.387C23, (C2xC4).250C24, Q8.D4:17C2, C22:C4.141D4, (C4xD4).68C22, C23.447(C2xD4), C4:Q8.101C22, (C2xQ8).43C23, (C4xQ8).65C22, C4.171(C4:D4), C2.15(D4oSD16), (C2xD4).389C23, (C2xD8).118C22, C23.36D4:8C2, C22.7(C4:D4), (C2xQ16).54C22, (C2xSD16).7C22, D4:C4.22C22, C42.6C22:8C2, (C22xC4).980C23, (C22xC8).179C22, C4.4D4.27C22, C22.510(C22xD4), C23.33C23:8C2, C23.38C23:9C2, Q8:C4.170C22, (C2xM4(2)).57C22, (C22xQ8).276C22, C42:C2.105C22, (C2xC4oD8).6C2, C4.160(C2xC4oD4), (C2xC4).470(C2xD4), C2.68(C2xC4:D4), (C2xQ8:C4):29C2, (C2xC8.C22):17C2, (C2xC4).281(C4oD4), (C2xC4:C4).584C22, (C2xC4oD4).122C22, SmallGroup(128,1778)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C42.19C23
C1C2C4C2xC4C22xC4C2xC4oD4C23.33C23 — C42.19C23
C1C2C2xC4 — C42.19C23
C1C22C42:C2 — C42.19C23
C1C2C2C2xC4 — C42.19C23

Generators and relations for C42.19C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=b2, e2=a2, ab=ba, cac-1=a-1, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >

Subgroups: 436 in 236 conjugacy classes, 100 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, D4:C4, Q8:C4, C4:C8, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C22:Q8, C22.D4, C4.4D4, C4:Q8, C22xC8, C2xM4(2), C2xD8, C2xSD16, C2xSD16, C2xQ16, C2xQ16, C4oD8, C8.C22, C22xQ8, C2xC4oD4, C2xQ8:C4, C23.36D4, C42.6C22, D4.D4, C4:2Q16, D4.2D4, Q8.D4, C23.33C23, C23.38C23, C2xC4oD8, C2xC8.C22, C42.19C23
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, C2xC4:D4, D4oSD16, Q8oD8, C42.19C23

Smallest permutation representation of C42.19C23
On 64 points
Generators in S64
(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)
(1 23 57 12)(2 24 58 9)(3 21 59 10)(4 22 60 11)(5 47 40 27)(6 48 37 28)(7 45 38 25)(8 46 39 26)(13 36 19 53)(14 33 20 54)(15 34 17 55)(16 35 18 56)(29 61 44 52)(30 62 41 49)(31 63 42 50)(32 64 43 51)
(1 53 57 36)(2 56 58 35)(3 55 59 34)(4 54 60 33)(5 30 40 41)(6 29 37 44)(7 32 38 43)(8 31 39 42)(9 16 24 18)(10 15 21 17)(11 14 22 20)(12 13 23 19)(25 64 45 51)(26 63 46 50)(27 62 47 49)(28 61 48 52)
(1 37)(2 38)(3 39)(4 40)(5 60)(6 57)(7 58)(8 59)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)(33 49)(34 50)(35 51)(36 52)(53 61)(54 62)(55 63)(56 64)
(1 33 3 35)(2 55 4 53)(5 52 7 50)(6 62 8 64)(9 17 11 19)(10 16 12 14)(13 24 15 22)(18 23 20 21)(25 42 27 44)(26 32 28 30)(29 45 31 47)(34 60 36 58)(37 49 39 51)(38 63 40 61)(41 46 43 48)(54 59 56 57)

G:=sub<Sym(64)| (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), (1,23,57,12)(2,24,58,9)(3,21,59,10)(4,22,60,11)(5,47,40,27)(6,48,37,28)(7,45,38,25)(8,46,39,26)(13,36,19,53)(14,33,20,54)(15,34,17,55)(16,35,18,56)(29,61,44,52)(30,62,41,49)(31,63,42,50)(32,64,43,51), (1,53,57,36)(2,56,58,35)(3,55,59,34)(4,54,60,33)(5,30,40,41)(6,29,37,44)(7,32,38,43)(8,31,39,42)(9,16,24,18)(10,15,21,17)(11,14,22,20)(12,13,23,19)(25,64,45,51)(26,63,46,50)(27,62,47,49)(28,61,48,52), (1,37)(2,38)(3,39)(4,40)(5,60)(6,57)(7,58)(8,59)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45)(33,49)(34,50)(35,51)(36,52)(53,61)(54,62)(55,63)(56,64), (1,33,3,35)(2,55,4,53)(5,52,7,50)(6,62,8,64)(9,17,11,19)(10,16,12,14)(13,24,15,22)(18,23,20,21)(25,42,27,44)(26,32,28,30)(29,45,31,47)(34,60,36,58)(37,49,39,51)(38,63,40,61)(41,46,43,48)(54,59,56,57)>;

G:=Group( (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), (1,23,57,12)(2,24,58,9)(3,21,59,10)(4,22,60,11)(5,47,40,27)(6,48,37,28)(7,45,38,25)(8,46,39,26)(13,36,19,53)(14,33,20,54)(15,34,17,55)(16,35,18,56)(29,61,44,52)(30,62,41,49)(31,63,42,50)(32,64,43,51), (1,53,57,36)(2,56,58,35)(3,55,59,34)(4,54,60,33)(5,30,40,41)(6,29,37,44)(7,32,38,43)(8,31,39,42)(9,16,24,18)(10,15,21,17)(11,14,22,20)(12,13,23,19)(25,64,45,51)(26,63,46,50)(27,62,47,49)(28,61,48,52), (1,37)(2,38)(3,39)(4,40)(5,60)(6,57)(7,58)(8,59)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45)(33,49)(34,50)(35,51)(36,52)(53,61)(54,62)(55,63)(56,64), (1,33,3,35)(2,55,4,53)(5,52,7,50)(6,62,8,64)(9,17,11,19)(10,16,12,14)(13,24,15,22)(18,23,20,21)(25,42,27,44)(26,32,28,30)(29,45,31,47)(34,60,36,58)(37,49,39,51)(38,63,40,61)(41,46,43,48)(54,59,56,57) );

G=PermutationGroup([[(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)], [(1,23,57,12),(2,24,58,9),(3,21,59,10),(4,22,60,11),(5,47,40,27),(6,48,37,28),(7,45,38,25),(8,46,39,26),(13,36,19,53),(14,33,20,54),(15,34,17,55),(16,35,18,56),(29,61,44,52),(30,62,41,49),(31,63,42,50),(32,64,43,51)], [(1,53,57,36),(2,56,58,35),(3,55,59,34),(4,54,60,33),(5,30,40,41),(6,29,37,44),(7,32,38,43),(8,31,39,42),(9,16,24,18),(10,15,21,17),(11,14,22,20),(12,13,23,19),(25,64,45,51),(26,63,46,50),(27,62,47,49),(28,61,48,52)], [(1,37),(2,38),(3,39),(4,40),(5,60),(6,57),(7,58),(8,59),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,42),(18,43),(19,44),(20,41),(21,46),(22,47),(23,48),(24,45),(33,49),(34,50),(35,51),(36,52),(53,61),(54,62),(55,63),(56,64)], [(1,33,3,35),(2,55,4,53),(5,52,7,50),(6,62,8,64),(9,17,11,19),(10,16,12,14),(13,24,15,22),(18,23,20,21),(25,42,27,44),(26,32,28,30),(29,45,31,47),(34,60,36,58),(37,49,39,51),(38,63,40,61),(41,46,43,48),(54,59,56,57)]])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4N4O4P4Q8A8B8C8D8E8F
order12222222244444···4444888888
size11112244822224···4888444488

32 irreducible representations

dim111111111111222244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4oD4D4oSD16Q8oD8
kernelC42.19C23C2xQ8:C4C23.36D4C42.6C22D4.D4C4:2Q16D4.2D4Q8.D4C23.33C23C23.38C23C2xC4oD8C2xC8.C22C22:C4C4:C4C4oD4C2xC4C2C2
# reps111122221111224422

Matrix representation of C42.19C23 in GL6(F17)

9130000
1280000
0050120
0005012
00120120
00012012
,
100000
010000
000100
0016000
000001
0000160
,
100000
13160000
0000160
000001
001000
0001600
,
100000
010000
003300
0031400
000033
0000314
,
9130000
1280000
005050
000505
0050120
0005012

G:=sub<GL(6,GF(17))| [9,12,0,0,0,0,13,8,0,0,0,0,0,0,5,0,12,0,0,0,0,5,0,12,0,0,12,0,12,0,0,0,0,12,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,3,3,0,0,0,0,3,14],[9,12,0,0,0,0,13,8,0,0,0,0,0,0,5,0,5,0,0,0,0,5,0,5,0,0,5,0,12,0,0,0,0,5,0,12] >;

C42.19C23 in GAP, Magma, Sage, TeX

C_4^2._{19}C_2^3
% in TeX

G:=Group("C4^2.19C2^3");
// GroupNames label

G:=SmallGroup(128,1778);
// by ID

G=gap.SmallGroup(128,1778);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,248,4037,1027,124]);
// Polycyclic

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

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