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G = C23.22D8order 128 = 27

1st non-split extension by C23 of D8 acting via D8/C8=C2

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

Aliases: C23.22D8, C24.134D4, C23.12Q16, C8:7(C22:C4), (C22xC8):14C4, (C2xC8).347D4, C2.1(C8:7D4), (C23xC8).13C2, C22.32(C2xD8), C22:2(C2.D8), (C22xC4).90Q8, C23.68(C4:C4), C22.4Q16:4C2, (C22xC4).545D4, C23.745(C2xD4), C4.70(C22:Q8), C22.25(C2xQ16), C2.1(C8.18D4), C22.44(C4oD8), C4.53(C42:C2), C23.7Q8.7C2, (C23xC4).671C22, (C22xC8).523C22, C22.110(C4:D4), (C22xC4).1330C23, C2.8(C23.25D4), C2.17(C23.7Q8), (C2xC2.D8):1C2, C2.6(C2xC2.D8), (C2xC4).84(C4:C4), (C2xC8).209(C2xC4), C4.87(C2xC22:C4), C22.91(C2xC4:C4), (C2xC4).188(C2xQ8), (C2xC4).1319(C2xD4), (C2xC4:C4).37C22, (C2xC4).552(C4oD4), (C22xC4).481(C2xC4), (C2xC4).528(C22xC4), SmallGroup(128,540)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C23.22D8
C1C2C22C23C22xC4C23xC4C23xC8 — C23.22D8
C1C2C2xC4 — C23.22D8
C1C23C23xC4 — C23.22D8
C1C2C2C22xC4 — C23.22D8

Generators and relations for C23.22D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 316 in 168 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C23, C23, C22:C4, C4:C4, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C24, C2.C42, C2.D8, C2xC22:C4, C2xC4:C4, C22xC8, C22xC8, C22xC8, C23xC4, C22.4Q16, C23.7Q8, C2xC2.D8, C23xC8, C23.22D8
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C22:C4, C4:C4, D8, Q16, C22xC4, C2xD4, C2xQ8, C4oD4, C2.D8, C2xC22:C4, C2xC4:C4, C42:C2, C4:D4, C22:Q8, C2xD8, C2xQ16, C4oD8, C23.7Q8, C2xC2.D8, C23.25D4, C8:7D4, C8.18D4, C23.22D8

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···122222···28···82···2

44 irreducible representations

dim11111122222222
type+++++++-++-
imageC1C2C2C2C2C4D4D4Q8D4C4oD4D8Q16C4oD8
kernelC23.22D8C22.4Q16C23.7Q8C2xC2.D8C23xC8C22xC8C2xC8C22xC4C22xC4C24C2xC4C23C23C22
# reps12221841214448

Matrix representation of C23.22D8 in GL5(F17)

10000
01000
00100
00010
000016
,
10000
01000
00100
000160
000016
,
160000
01000
00100
00010
00001
,
10000
031400
03300
00020
00009
,
40000
061300
0131100
00009
00020

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,2,0,0,0,0,0,9],[4,0,0,0,0,0,6,13,0,0,0,13,11,0,0,0,0,0,0,2,0,0,0,9,0] >;

C23.22D8 in GAP, Magma, Sage, TeX

C_2^3._{22}D_8
% in TeX

G:=Group("C2^3.22D8");
// GroupNames label

G:=SmallGroup(128,540);
// by ID

G=gap.SmallGroup(128,540);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2019,248]);
// Polycyclic

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

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