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G = C23.23D8order 128 = 27

2nd non-split extension by C23 of D8 acting via D8/C8=C2

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

Aliases: C23.23D8, C24.136D4, C23.26SD16, (C2xC8):36D4, (C23xC8):2C2, C4:D4:12C4, C4.113(C4xD4), C4.90C22wrC2, C2.3(C8:8D4), C2.2(C8:7D4), C22.40(C2xD8), C23.773(C2xD4), (C22xC4).551D4, C23.7Q8:6C2, C22:2(D4:C4), C22.4Q16:15C2, C22.60(C4oD8), C22.61(C2xSD16), (C22xC8).482C22, (C23xC4).674C22, (C22xD4).24C22, C22.119(C4:D4), C23.122(C22:C4), (C22xC4).1369C23, C4.85(C22.D4), C2.32(C23.23D4), C2.27(C23.24D4), C4:C4.72(C2xC4), (C2xD4:C4):6C2, (C2xD4).81(C2xC4), (C2xC4:D4).8C2, C2.21(C2xD4:C4), (C2xC4).1332(C2xD4), (C2xC4:C4).61C22, (C2xC4).566(C4oD4), (C2xC4).387(C22xC4), (C22xC4).405(C2xC4), (C2xC4).194(C22:C4), C22.268(C2xC22:C4), SmallGroup(128,625)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C23.23D8
C1C2C22C23C22xC4C23xC4C23xC8 — C23.23D8
C1C2C2xC4 — C23.23D8
C1C23C23xC4 — C23.23D8
C1C2C2C22xC4 — C23.23D8

Generators and relations for C23.23D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, eae=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 484 in 216 conjugacy classes, 72 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C24, C2.C42, D4:C4, C2xC22:C4, C2xC4:C4, C2xC4:C4, C4:D4, C4:D4, C22xC8, C22xC8, C23xC4, C22xD4, C22xD4, C22.4Q16, C23.7Q8, C2xD4:C4, C2xC4:D4, C23xC8, C23.23D8
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, D8, SD16, C22xC4, C2xD4, C4oD4, D4:C4, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C2xD8, C2xSD16, C4oD8, C23.23D4, C2xD4:C4, C23.24D4, C8:8D4, C8:7D4, C23.23D8

Smallest permutation representation of C23.23D8
On 64 points
Generators in S64
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 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 29)(2 6)(3 27)(5 25)(7 31)(9 53)(10 23)(11 51)(12 21)(13 49)(14 19)(15 55)(16 17)(18 45)(20 43)(22 41)(24 47)(28 32)(33 63)(34 38)(35 61)(37 59)(39 57)(42 50)(44 56)(46 54)(48 52)(58 62)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N8A···8P
order12···22222224···44···48···8
size11···12222882···28···82···2

44 irreducible representations

dim11111112222222
type++++++++++
imageC1C2C2C2C2C2C4D4D4D4C4oD4D8SD16C4oD8
kernelC23.23D8C22.4Q16C23.7Q8C2xD4:C4C2xC4:D4C23xC8C4:D4C2xC8C22xC4C24C2xC4C23C23C22
# reps12121184314448

Matrix representation of C23.23D8 in GL5(F17)

160000
04900
041300
00010
00001
,
10000
016000
001600
00010
00001
,
160000
01000
00100
000160
000016
,
40000
001100
031100
00007
00057
,
160000
016200
00100
000162
00001

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,4,4,0,0,0,9,13,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,3,0,0,0,11,11,0,0,0,0,0,0,5,0,0,0,7,7],[16,0,0,0,0,0,16,0,0,0,0,2,1,0,0,0,0,0,16,0,0,0,0,2,1] >;

C23.23D8 in GAP, Magma, Sage, TeX

C_2^3._{23}D_8
% in TeX

G:=Group("C2^3.23D8");
// GroupNames label

G:=SmallGroup(128,625);
// by ID

G=gap.SmallGroup(128,625);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,248]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,e*a*e=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations

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