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G = C23.12D8order 128 = 27

5th non-split extension by C23 of D8 acting via D8/C4=C22

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

Aliases: C23.12D8, C24.87D4, (C2×C8).53D4, C22.89(C2×D8), C2.16(C87D4), (C22×C4).160D4, C23.930(C2×D4), C22.4Q1629C2, C4.54(C4.4D4), C2.16(C8.D4), C4.18(C422C2), C22.122(C4○D8), (C22×C8).115C22, (C23×C4).275C22, C2.8(C22.D8), C23.7Q8.20C2, C22.251(C4⋊D4), (C22×C4).1464C23, C4.110(C22.D4), C2.11(C23.20D4), C2.10(C23.11D4), C22.140(C8.C22), C22.120(C22.D4), (C2×C2.D8)⋊10C2, (C2×C4).1373(C2×D4), (C2×C22⋊C8).28C2, (C2×C4).626(C4○D4), (C2×C4⋊C4).149C22, SmallGroup(128,807)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C23.12D8
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.7Q8 — C23.12D8
C1C2C22×C4 — C23.12D8
C1C23C23×C4 — C23.12D8
C1C2C2C22×C4 — C23.12D8

Generators and relations for C23.12D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=b, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 288 in 128 conjugacy classes, 48 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×5], C22 [×3], C22 [×4], C22 [×10], C8 [×3], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], C23, C23 [×2], C23 [×6], C22⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×5], C22×C4 [×2], C22×C4 [×10], C24, C2.C42 [×2], C22⋊C8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C23×C4, C22.4Q16, C22.4Q16 [×2], C23.7Q8 [×2], C2×C22⋊C8, C2×C2.D8, C23.12D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C2×D8, C4○D8, C8.C22 [×2], C23.11D4, C87D4, C8.D4, C22.D8 [×2], C23.20D4 [×2], C23.12D8

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N8A···8H
order12···2224444444···48···8
size11···1442222448···84···4

32 irreducible representations

dim111112222224
type+++++++++-
imageC1C2C2C2C2D4D4D4C4○D4D8C4○D8C8.C22
kernelC23.12D8C22.4Q16C23.7Q8C2×C22⋊C8C2×C2.D8C2×C8C22×C4C24C2×C4C23C22C22
# reps1321121110442

Matrix representation of C23.12D8 in GL6(𝔽17)

1600000
0160000
001000
0071600
000010
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
001000
000100
0000160
0000016
,
200000
090000
0010200
0010700
000004
000040
,
090000
200000
0011900
0011600
000040
0000013

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[2,0,0,0,0,0,0,9,0,0,0,0,0,0,10,10,0,0,0,0,2,7,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,11,11,0,0,0,0,9,6,0,0,0,0,0,0,4,0,0,0,0,0,0,13] >;

C23.12D8 in GAP, Magma, Sage, TeX

C_2^3._{12}D_8
% in TeX

G:=Group("C2^3.12D8");
// GroupNames label

G:=SmallGroup(128,807);
// by ID

G=gap.SmallGroup(128,807);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,718,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,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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