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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, (C2×C8)⋊36D4, (C23×C8)⋊2C2, C4⋊D412C4, C4.113(C4×D4), C4.90C22≀C2, C2.3(C88D4), C2.2(C87D4), C22.40(C2×D8), C23.773(C2×D4), (C22×C4).551D4, C23.7Q86C2, C222(D4⋊C4), C22.4Q1615C2, C22.60(C4○D8), C22.61(C2×SD16), (C22×C8).482C22, (C23×C4).674C22, (C22×D4).24C22, C22.119(C4⋊D4), C23.122(C22⋊C4), (C22×C4).1369C23, C4.85(C22.D4), C2.32(C23.23D4), C2.27(C23.24D4), C4⋊C4.72(C2×C4), (C2×D4⋊C4)⋊6C2, (C2×D4).81(C2×C4), (C2×C4⋊D4).8C2, C2.21(C2×D4⋊C4), (C2×C4).1332(C2×D4), (C2×C4⋊C4).61C22, (C2×C4).566(C4○D4), (C2×C4).387(C22×C4), (C22×C4).405(C2×C4), (C2×C4).194(C22⋊C4), C22.268(C2×C22⋊C4), SmallGroup(128,625)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.23D8
C1C2C22C23C22×C4C23×C4C23×C8 — C23.23D8
C1C2C2×C4 — C23.23D8
C1C23C23×C4 — C23.23D8
C1C2C2C22×C4 — 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 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×6], C22 [×3], C22 [×8], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], C23, C23 [×6], C23 [×12], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C24, C24, C2.C42, D4⋊C4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C22×C8 [×6], C23×C4, C22×D4, C22×D4, C22.4Q16 [×2], C23.7Q8, C2×D4⋊C4 [×2], C2×C4⋊D4, C23×C8, C23.23D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×D8, C2×SD16, C4○D8 [×2], C23.23D4, C2×D4⋊C4, C23.24D4, C88D4 [×2], C87D4 [×2], C23.23D8

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

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

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

G=PermutationGroup([(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,54),(18,55),(19,56),(20,49),(21,50),(22,51),(23,52),(24,53),(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),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,48),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,49),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(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)], [(1,56),(2,6),(3,54),(5,52),(7,50),(9,18),(10,27),(11,24),(12,25),(13,22),(14,31),(15,20),(16,29),(17,48),(19,46),(21,44),(23,42),(26,41),(28,47),(30,45),(32,43),(33,63),(34,38),(35,61),(37,59),(39,57),(51,55),(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++++++++++
imageC1C2C2C2C2C2C4D4D4D4C4○D4D8SD16C4○D8
kernelC23.23D8C22.4Q16C23.7Q8C2×D4⋊C4C2×C4⋊D4C23×C8C4⋊D4C2×C8C22×C4C24C2×C4C23C23C22
# 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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