C4oD4:C12 - GroupNames

G = C₄○D₄⋊C₁₂ order 192 = 2⁶·3

The semidirect product of C₄○D₄ and C₁₂ acting via C₁₂/C₂=C₆

Aliases: C₄○D₄⋊C₁₂, C₄⋊C₄.₁A₄, C₄.A₄⋊₅C₄, C₄.₁(C₄×A₄), (C₄×Q₈).₁C₆, Q₈.₃(C₂×C₁₂), C₂.₂(D₄.A₄), C₂.₁(Q₈.A₄), C₂³.₃₃C₂³⋊C₃, SL₂(𝔽₃)⋊₇(C₂×C₄), (C₄×SL₂(𝔽₃))⋊₄C₂, C₂².₁₆(C₂²×A₄), (C₂×SL₂(𝔽₃)).₂₆C₂², C₂.₈(C₂×C₄×A₄), (C₂×C₄).₃(C₂×A₄), (C₂×C₄.A₄).₇C₂, (C₂×C₄○D₄).₁C₆, (C₂×Q₈).₃₇(C₂×C₆), SmallGroup(192,999)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — Q₈ — C₄○D₄⋊C₁₂

Chief series

C₁ — C₂ — Q₈ — C₂×Q₈ — C₂×SL₂(𝔽₃) — C₂×C₄.A₄ — C₄○D₄⋊C₁₂

Lower central

Q₈ — C₄○D₄⋊C₁₂

Upper central

C₁ — C₂² — C₄⋊C₄

Generators and relations for C₄○D₄⋊C₁₂
G = < a,b,c,d | a⁴=c²=d¹²=1, b²=a², dcd^-1=ab=ba, ac=ca, dad^-1=a^-1, cbc=a²b, dbd^-1=a^-1bc >

Subgroups: 253 in 84 conjugacy classes, 27 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₁₂, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, SL₂(𝔽₃), C₂×C₁₂, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂×C₄○D₄, C₃×C₄⋊C₄, C₂×SL₂(𝔽₃), C₄.A₄, C₂³.₃₃C₂³, C₄×SL₂(𝔽₃), C₂×C₄.A₄, C₄○D₄⋊C₁₂
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, C₁₂, A₄, C₂×C₆, C₂×C₁₂, C₂×A₄, C₄×A₄, C₂²×A₄, C₂×C₄×A₄, Q₈.A₄, D₄.A₄, C₄○D₄⋊C₁₂

Smallest permutation representation of C₄○D₄⋊C₁₂
►On 64 points

Generators in S₆₄

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

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

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

(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)

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	4A	···	4F	4G	···	4L	6A	···	6F	12A	···	12L
order	1	2	2	2	2	2	3	3	4	···	4	4	···	4	6	···	6	12	···	12
size	1	1	1	1	6	6	4	4	2	···	2	6	···	6	4	···	4	8	···	8

38 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	4	4	4	4
type	+	+	+						+	+		+		-
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	A₄	C₂×A₄	C₄×A₄	Q₈.A₄	Q₈.A₄	D₄.A₄	D₄.A₄
kernel	C₄○D₄⋊C₁₂	C₄×SL₂(𝔽₃)	C₂×C₄.A₄	C₂³.₃₃C₂³	C₄.A₄	C₄×Q₈	C₂×C₄○D₄	C₄○D₄	C₄⋊C₄	C₂×C₄	C₄	C₂	C₂	C₂	C₂
# reps	1	2	1	2	4	4	2	8	1	3	4	1	2	1	2

Matrix representation of C₄○D₄⋊C₁₂ ►in GL₇(𝔽₁₃)

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	5	0	0	0
0	0	0	0	5	0	0
0	0	0	0	0	8	0
0	0	0	0	0	0	8

0	1	0	0	0	0	0
1	0	0	0	0	0	0
12	12	12	0	0	0	0
0	0	0	0	12	0	0
0	0	0	1	0	0	0
0	0	0	0	0	0	12
0	0	0	0	0	1	0

12	12	12	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	6	2	0	0
0	0	0	2	7	0	0
0	0	0	0	0	7	11
0	0	0	0	0	11	6

5	0	0	0	0	0	0
0	0	5	0	0	0	0
8	8	8	0	0	0	0
0	0	0	0	0	1	0
0	0	0	0	0	10	9
0	0	0	1	0	0	0
0	0	0	10	9	0	0

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8],[0,1,12,0,0,0,0,1,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,6,2,0,0,0,0,0,2,7,0,0,0,0,0,0,0,7,11,0,0,0,0,0,11,6],[5,0,8,0,0,0,0,0,0,8,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,9,0,0,0,1,10,0,0,0,0,0,0,9,0,0] >;

C₄○D₄⋊C₁₂ in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes C_{12}

% in TeX

G:=Group("C4oD4:C12");

// GroupNames label

G:=SmallGroup(192,999);

// by ID

G=gap.SmallGroup(192,999);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,672,1373,92,438,172,775,285,124]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^2=d^12=1,b^2=a^2,d*c*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^2*b,d*b*d^-1=a^-1*b*c>;

// generators/relations

G = C4○D4⋊C12 order 192 = 26·3

The semidirect product of C4○D4 and C12 acting via C12/C2=C6

G = C₄○D₄⋊C₁₂ order 192 = 2⁶·3

The semidirect product of C₄○D₄ and C₁₂ acting via C₁₂/C₂=C₆